1 Introduction↩︎

It was already known to Dehn that any finitely presented group can be realized as the fundamental group of a closed manifold of any dimension at least four, but this is not the case for three-manifolds, e.g., the group \(\mathbb{Z}^4\) is not the fundamental group of any closed three-manifold. This makes the class of three-dimensional manifolds special and we may expect that their fundamental groups enjoy specific properties which characterize them among finitely generated groups.

From the point of view of geometric group theory, one tries to understand the properties of a group by studying the different actions it admits on metric spaces. For the action of the group \(G\) on the geodesic metric space \(X\) to properly reflect the properties of \(G\), we require that it is geometric: the group \(G\) acts by isometries (the action is distance-preserving), properly discontinuously (for any compact subsets \(K\) and \(L\) of \(X\), at most finitely many elements \(g\) of \(G\) will satisfy \(g(K)\cap L \ne\emptyset\)) and cocompactly (the orbit space \(X/G\) is compact). By identifying \(G\) with the orbit \(Go\) of a point \(o\in X\) and by pulling back the induced metric, we obtain a metric on \(G\). Changing the orbit or the metric space \(X\) gives rise to new metrics. Thus we get a metric structure on \(G\) which is coarsely defined in the following sense.

A quasi-isometry between metric spaces \(X\) and \(Y\) is a coarsely bi-Lipschitz coarsely surjective map \(\varphi:X\to Y\), i.e., there are constants \(\lambda>1\) and \(c>0\) such that:

• (quasi-isometric embedding) for all \(x,x'\in X\), the two inequalities \[\frac{1}{\lambda}d_X(x,x') - c \le d_Y(\varphi(x),\varphi(x'))\le \lambda\, d_X(x,x') + c\] hold and

• (quasi-surjectivity) the \(c\)-neighborhood of the image \(f(X)\) covers \(Y\).

This defines in fact an equivalence relation on (separable) metric spaces. The Švarc–Milnor lemma asserts that there is only one geometric action of a group on a proper geodesic metric space up to quasi-isometry [1]: we equip \(G\) with a reference metric induced by identifying \(G\) with an orbit \(G o\) under a geometric action on some proper geodesic space \(Y\) (usually one takes its left action on one of its locally finite Cayley graphs).

Let \(X\) be a proper geodesic metric space. Let \(G\) act properly discontinuously and cocompactly on \(X\) by isometries. Then \(G\) is finitely generated and, for any \(x_0\in X\), the map \(g\mapsto g.\;x_0\) is a quasi-isometry.

Thus we have a coarsely well-defined metric on \(G\), i.e., defined up to quasi-isometry, coming from its geometric actions. We are then naturally led to ask whether or not a property of a finitely generated group is invariant under quasi-isometries or, equivalently, whether or not a class of groups is quasi-isometrically rigid: a class of groups \(\mathcal{C}\) is quasi-isometrically rigid if any group quasi-isometric to a group in \(\mathcal{C}\) has a finite index subgroup in \(\mathcal{C}\).

Since the pioneering works of Stallings and Gromov, diverse classes of groups have been proved to be quasi-isometrically rigid. We should mention free groups [2], nilpotent groups [3], Abelian groups [3]–[7] and word hyperbolic groups. A more thorough overview of these results will be given in §2.1.

In contrast with higher dimension (every finitely presented group is the fundamental group of a compact \(4\)-manifold), fundamental groups of low-dimensional compact manifolds have many restrictive properties and their quasi-isometric rigidity is a challenging question that has already led to many interesting developments. In this paper we focus on compact three-manifolds but the reader should be aware that the quasi-isometric rigidity of surface groups follows from [8], [9] and results we have already mentioned (see §2.1 for more details). Our main theorem completes the work of many people whose combined results can be fairly accurately summarized in the following statement; see §2.1 for more details about these results and §2.1 and §2.2 for the definition of irreducible three-manifolds.

As we will see in §4.2, it is relatively easy, using the work of [10], to remove the assumption that \(M\) is irreducible. Thus extending Theorem 1 to manifolds with negative Euler characteristic completely settles the question. This is the purpose of the present paper which leads to the following statement:

Notice that \(\pi_1(M)\) and \(\pi_1(N)\) need not be commensurable, i.e., may have no isomorphic finite index subgroups. For example, consider closed quotients of \(\mathbb{H}^2\times \mathbb{E}^1\) and \(\widetilde{SL_2(\mathbb{R})}\): since \(\mathbb{H}^2\times \mathbb{E}^1\) and \(\widetilde{SL_2}(\mathbb{R})\) are quasi-isometric, the fundamental groups of all those quotients are quasi-isometric by Švarc–Milnor lemma, but \(\mathbb{H}^2\times \mathbb{E}^1\) and \(\widetilde{SL_2(\mathbb{R})}\) have no isomorphic lattices, see [11] and [12]. Non-geometric examples can also be produced, using [13]–[15]. Nevertheless, it follows from the present work that, for a non-geometric irreducible manifold or a hyperbolic manifold with non-empty boundary, the fundamental groups of the pieces obtained after cutting the manifold along compressing discs, essential tori and annuli are well-defined up to commensurability, cf. Theorem 65.

Notice also that Theorem 2 involves a slight upgrade of Theorem 1 to go from a short exact sequence to a finite index subgroup. This comes from the following statement, interesting in its own right:

Two groups \(G\) and \(Q\) are commensurable if there are subgroups \(G'<G\) and \(Q'<Q\) of finite index such that \(G'\) is isomorphic to \(Q'\).

One of our main inputs deals with finitely generated Kleinian groups, i.e., discrete subgroups of \(\mathbb{P}SL_2(\mathbb{C})\). We provide a new proof of the quasi-isometric rigidity of convex-cocompact Kleinian groups [16], [17] which holds for all (finitely generated) Kleinian groups, leading to:

To add a little perspective to this introduction let us remark that quasi-isometric groups may also be fairly different from one another. For instance, simplicity, linearity and residual finiteness [18], Kazhdan’s property (T) ([19] and [20]) and the Haagerup property [21] are not virtually invariant under quasi-isometries. We refer the interested reader to [22], [23] and [20] for more properties that are known to be invariant or not under quasi-isometries.

When considering three-manifold groups from the point of view of geometric group theory it is also natural to wonder about their classification up to quasi-isometry. This question has been the subject of extensive work of Behrstock and Neumann [24], [25] who give a nearly complete answer.

Outline of the paper. The proofs of Theorems 2 and 4 can be decomposed into three steps which are similar for both proofs. Let \(Q=\pi_1(M)\) be either the fundamental group of a compact three-manifold (for Theorem 2) or a compact hyperbolic three-manifold (for Theorem 4) and let \(G\) be a finitely generated group quasi-isometric to \(Q\).

In the first step, we simultaneously split \(G\) and \(Q\) to get graphs of groups \[{\mathcal{G}}=(\Gamma_G,\{G_v\},\{G_e\}, G_e\hookrightarrow G_{t(e)})and{\mathcal{Q}}=(\Gamma_Q,\{Q_v\},\{Q_e\}, Q_e\hookrightarrow Q_{t(e)})\] (see §4.1 for definitions and notations) with quasi-isometric vertex groups. The splittings of \(Q\) come from topological splittings of \(M\) and the vertex groups \(Q_v\) are fundamental groups of hyperbolic three-manifolds and of compact three-manifolds with zero Euler characteristic in the proof of Theorem 2, while in the proof of Theorem 4, they are fundamental groups of pared acylindrical hyperbolic three-manifolds and pared \(I\)-bundles.

The second step consists in showing that the classes to which the vertex groups \(Q_v\) belong are quasi-isometrically rigid. For Theorem 2, it is given by Theorems 4 and 1. For Theorem 4, it is the quasi-isometric rigidity of pared acylindrical Kleinian groups (Theorem 61) and pared \(I\)-bundles (Lemma 19). Applying these results to the vertex groups \(G_v\) we get finite index subgroups \(G'_v<G_v\) which are fundamental groups of compact three-manifolds \(M_v\).

In the third and last step we find a finite index subgroup \(G'<G\) whose intersection with each vertex group \(G_v\) is a finite index subgroup of \(G'_v\) and deduce that \(G'\) is the fundamental group of a compact three-manifold obtained by gluing together finite covers of the manifolds \(M_v\).

The arguments used in these steps are independent and for a better exposition we will study them in a different order. We will give more insights on these steps while we detail the plan of the paper.

First, in Section 2.1 we review results on quasi-isometric rigidity that are related to our topic. In particular, we recount the results that lead to Theorem 1. In Section 2.2 we provide background on three-manifolds and Kleinian groups. In particular, we show that a finitely generated Kleinian group is always isomorphic to a so-called geometrically finite Kleinian group with minimal parabolics, cf. Proposition 16. In Section 3 we introduce word hyperbolic and relatively hyperbolic groups and establish some facts that will be used later on. In Section 4 we establish the first step described above, i.e., we study characteristic splittings of \(Q\) and \(G\) and their quasi-isometric invariance. In Section 4.2, we introduce a maximal splitting along spheres and discs and its analog for groups [26] which are used in the proofs of both Theorems 2 and 4. Its quasi-isometric invariance has been established in [10]. In Section 4.3, we describe the characteristic torus decomposition and the induced splittings which are used in the proof of Theorem 2. The analog for groups and its quasi-isometric invariance are built on the work of Kapovich–Leeb [27]. In Section 4.4, we introduce the characteristic annulus decomposition which is used in the proof of Theorem 4. Its analog for groups is defined using the work of [28] on trees associated to the cut points and cut pairs of a continuum. The quasi-isometric invariance of the splitting thus produced is proved using the relation between the limit set of a Kleinian group and the characteristic submanifold of the underlying manifold [29], see also [30]. In Section 5 we set up the third step by building finite index subgroups of fundamental groups of graph of groups with prescribed intersections with edge and vertex groups. The main tool used here is Wise’s virtually special quotient theorem [31], see also [31]. In the last section, we conclude the proof of our main theorems. First we show Theorem 3 in Section 6.1 using an induction argument and hierarchies in groups quasi-isometric to three-manifold groups. In Section 6.2, we establish the second step of the proof of Theorem 4 by proving the quasi-isometric rigidity of pared \(I\)-bundles and pared acylindrical Kleinian groups. Lastly, in Sections 6.3 and 6.4, we proceed with the proof of Theorems 4 and 2 as explained above.

Acknowledgements.– We are grateful to Misha Kapovich for having brought this question to our attention. We also feel particularly indebted to Dani Wise for his help regarding the construction of finite index subgroups and for his enthusiastic support on this project. It is our pleasure to thank Daniel Groves and Jason Manning for fruitful discussions. We also thank Jack Button for pointing out a mistake in the introduction of a previous version and Chris Hruska for noticing that arguments from the proof of Theorem 3 can be used to remove the faithfulness assumption in the Bieberbach theorem (Theorem 58). We are grateful to the referees for their thorough reading of this work and their many suggestions that have improved the paper. This work was partially supported by the ANR project “GDSous/GSG” no. 12-BS01-0003-01.

2 Topology and geometry of three-manifolds↩︎

2.1 Quasi-isometric rigidity and geometric manifolds↩︎

To complete our introduction, we will now give more details on the quasi-isometric rigidity results that were mentioned earlier.

The work of Stallings on ends of groups [2] is a natural starting point. It leads to the quasi-isometric rigidity of virtually free groups, see [20]. A group \(G\) is said to virtually have a property if a finite index subgroup \(H\) of \(G\) has the said property.

Another class of groups for which the quasi-isometric rigidity has been established is the class of virtually nilpotent groups. The next result follows from Gromov’s polynomial growth theorem [3]:

Combining Theorem 6 with Bass-Guivarc’h formula for the polynomial growth of nilpotent groups [4], [5] and the work of Pansu on their asymptotic cones [6], one gets [20]:

For a proof avoiding the classification of groups of polynomial growth, see [7].

2.1.1 Surface groups↩︎

Next, we explain the quasi-isometric rigidity of surface groups.

To make good use of the Švarc–Milnor lemma, we want to put a convenient metric on a compact manifold. In dimension \(2\), the Poincaré-Koebe uniformization theorem provides us with a spherical, Euclidean or hyperbolic metric for any compact surface. For better consistency with the \(3\) dimensional case, let us put that statement in the perspective of Thurston’s model geometry.

A model geometry \((G,X)\) is a manifold \(X\) together with a Lie group \(G\) of diffeomorphisms of \(X\), such that:

(a) \(X\) is connected and simply connected;

(b) \(G\) acts transitively on \(X\), with compact point stabilizers;

(c) \(G\) is not contained in any larger group of diffeomorphisms of \(X\) with compact stabilizers of points; and

(d) there exists at least one closed manifold \(M\) modelled on \((G,X)\), meaning that there is a diffeomorphism from \(M\) to \(X/\Gamma\), where \(\Gamma\) is a discrete subgroup of \(G\) acting freely on \(X\).

More generally, if \(M\) is a compact manifold, we say that \(M\) is modelled on \((G,X)\) when there is a diffeomorphism from \(int(M)\) to \(X/\Gamma\). Then \(X/\Gamma\) defines a geometric structure and we say that a manifold is geometric if it has a geometric structure.

In dimension two there are three model geometries [32]: \(\mathbb{S}^2\), \(\mathbb{E}^2\) and \(\mathbb{H}^2\) together with their groups of isometries. Thus, the Poincaré-Koebe uniformization theorem says that all compact surfaces are geometric. Now the proof of Theorem 8 boils down to the rigidity of discrete subgroups of isometries of \(\mathbb{S}^2\), \(\mathbb{E}^2\) and \(\mathbb{H}^2\).

. As we have already explained, \(\pi_1(S)\) is isomorphic to a discrete subgroup \(\Gamma\) of \(Isom(X)\) with \(X=\mathbb{S}^2\), \(\mathbb{E}^2\) or \(\mathbb{H}^2\).

If \(X=\mathbb{S}^2\), \(G\) is finite and there is nothing to prove.

If \(X=\mathbb{E}^2\), \(\pi_1(S)\) is virtually Abelian of rank \(1\) or \(2\) depending on whether \(S\) has a boundary or not and the conclusion follows from Theorem 7; an alternate proof can be found in [33].

If \(X=\mathbb{H}^2\), \(\Gamma\) can be chosen to be a lattice of \(\mathbb{P}SL_2(\mathbb{R})\) —a Fuchsian group of finite area. If \(\Gamma\) is non uniform (equivalently if \(\partial S\neq\emptyset\)), it is a free group and the conclusion follows from Theorem 5. Thus we are only left with the case where \(G\) is quasi-isometric to a uniform lattice \(K<\mathbb{P}SL_2(\mathbb{R})\). Then \(K\) is word hyperbolic (see Definition 20) with boundary homeomorphic to \(S^1\), so \(G\) is as well. Therefore, \(G\) admits a uniform action on \(S^1\) and it follows from [9] or [8] that this action is conjugate to that of a cocompact Fuchsian group. Since \(G\) is a convergence group, the kernel of the action is finite. \(\rule{.55em}{.55em}\)

Combining Theorems 8 and 3, we get:

2.1.2 Three-manifold groups↩︎

For fundamental groups of three-manifolds, we summarized in the introduction the state of the art with the following statement:

A three-manifold is irreducible if every embedded sphere bounds a ball (the rest of the terminology used below is given in Section 2.2). As in the surface case, a first step in the proof consists in equipping three-manifolds with convenient metrics.

Thurston has shown [32] that there are eight three-dimensional model geometries \((G,X)\) which are \(\mathbb{S}^3\), \(\mathbb{E}^3\), \(\mathbb{H}^3\), \(\mathbb{S}^2\times\mathbb{E}^1\), \(\mathbb{H}^2\times\mathbb{E}^1\), \(Nil\), \(\widetilde{SL(2,\mathbb{R})}\), \(Sol\) together with their groups of isometries.

In general, three-manifolds are not geometric but the geometrization theorem (proved for Haken manifolds and stated as a conjecture in general by Thurston [34] and proved by Perel’man in general [35]–[37]) asserts that they can be decomposed into geometric pieces.

A geometric decomposition of a three-manifold \(M\) is a collection of essential tori \(T\) such that each component of \(M\setminus T\) has a geometric structure. The geometrization theorem precisely states that such a geometric decomposition always exists. We say that a geometric decomposition \(T\) is minimal if for any component \(T_1\) of \(T\), \(T\setminus T_1\) is not a geometric decomposition.

Let us now explain the main steps of the proof of Theorem 1, starting with geometric three-manifolds.

Proof. We have already seen that \(M\) is modelled on one of the homogeneous spaces \(X=\mathbb{S}^3, \mathbb{S}^2\times\mathbb{E}^1,\mathbb{E}^3, \mathbb{H}^2\times\mathbb{E}^1,\mathbb{H}^3, Nil, Sol, \widetilde{SL_2(\mathbb{R})}\).

When \(\pi_1(M)\) is finite, in particular when \(X=\mathbb{S}^3\) or when \(M\) has positive Euler characteristic, we take the trivial group considered as \(\pi_1(\mathbb{S}^3)\) for \(Q\). We now assume that \(\pi_1(M)\) is infinite so that its Euler characteristic is zero.

When \(X=\mathbb{S}^2\times\mathbb{E}^1\), then \(\pi_1(M)\) is virtually cyclic and the conclusion follows from Theorem 5.

When \(X=\mathbb{E}^3\), it is a special case of Theorem 7.

When \(X=Nil\), it follows from Theorem 6 and results of Mal’cev, Guivarc’h [5] and Jenkins [38], cf. [39].

When \(X=\mathbb{H}^3\), since \(M\) has zero Euler characteristic, its hyperbolic structure has finite volume (see [34] for example). The conclusion is due to Sullivan [40] and Cannon-Cooper [41] when \(\partial M=\emptyset\) and Schwartz [42] when \(\partial M\neq\emptyset\).

When \(X=Sol\), it has been proved by Eskin, Fisher and Whyte [43].

Finally, when \(X=\mathbb{H}^2\times\mathbb{E}^1\) and \(X=\widetilde{SL_2(\mathbb{R})}\), it is due to Rieffel [44], see also [27]. Note that these two geometries are quasi-isometric. \(\rule{.55em}{.55em}\)

Notice that geometric three-manifolds with negative Euler characteristic are hyperbolic, i.e., modeled on \(\mathbb{H}^3\).

From a strong invariance of the geometric decomposition under quasi-isometry, Kapovich–Leeb [27] deduce the quasi-isometric rigidity of fundamental groups of irreducible non-geometric three-manifolds with zero Euler characteristic. We will give more details in §4.3 since we will use this invariance to prove Theorem 2.

By the geometrization theorem (Theorem 10), any non-geometric compact three-manifold is Haken (see the next section for a definition). Thus we get Theorem 1 simply by combining Theorems 11 and 12.

2.2 Three-manifolds and groups↩︎

In this section and the next one, we will review some classical definitions and results that are used in this paper. We start with three-manifold topology, basic references include [45], [46].

2.2.1 Three-manifold topology↩︎

Let \(M\) be a compact orientable three-manifold with boundary or not. An embedded surface \((S,\partial S)\to (M,\partial M)\) is incompressible if the inclusion \(i:S\to M\) gives rise to an injective morphism \(i_*:\pi_1(S,x)\to\pi_1(M,x)\). The double of a manifold \(M\) with boundary is the union of \(M\) and of a copy of itself glued along its boundary. An embedded surface \(S\) in \(M\) is boundary incompressible if its double is incompressible in the double of \(M\). A surface \(S\) in \(M\) is non-peripheral if it is properly embedded, i.e., \(S\cap\partial M=\partial S\), and if the inclusion \(i:S\to M\) is not homotopic relative to \(\partial S\) to a map \(f:S\to M\) such that \(f(S)\subset\partial M\). A surface \(S\) is essential if it is properly embedded, two-sided, incompressible, boundary incompressible, non-peripheral and does not bound a \(3\)-ball. An essential disc is also called a compression disc.

The manifold \(M\) is irreducible if it contains no essential sphere, equivalently every embedded sphere bounds a ball. We say that \(M\) is boundary irreducible or \(\partial\)-irreducible if each component of \(\partial M\) is incompressible. By results of Kneser and Stallings, \(M\) is irreducible and boundary irreducible if and only if \(\pi_1(M)\) is one-ended.

The irreducible manifold \(M\) is Haken if it contains an essential surface. If \(\partial M\neq\emptyset\) then \(M\) is necessarily Haken, see [46].

An irreducible three-manifold \(M\) is atoroidal if every subgroup of \(\pi_1(M)\) isomorphic to \(\mathbb{Z}\oplus\mathbb{Z}\) is conjugate to a subgroup of the fundamental group of a boundary component. As we will see in the next section this property characterizes hyperbolic manifolds. An acylindrical compact manifold is atoroidal, has incompressible boundary and no essential annuli.

A Seifert manifold is a compact three-manifold which admits a foliation by circles. Even though this is not the classical definition, it is equivalent to it [47]. A graph manifold is a compact irreducible \(\partial\)-irreducible three-manifold with no atoroidal pieces in its torus decomposition, see §4.3.

A compact pared manifold \((M,P)\) is given by a compact irreducible three-manifold \(M\) non-homeomorphic to a solid torus nor a thickened torus together with a paring \(P\subset\partial M\) which is a finite collection of pairwise disjoint incompressible annuli and tori satisfying:

1. every Abelian, non cyclic subgroup of \(\pi_1(M)\) is conjugate to a subgroup of the fundamental group of a component of \(P\) and

2. any incompressible cylinder \((C,\partial C)\subset (M,P)\) can be homotoped relatively to its boundary into \(P\).

Notice that by definition only atoroidal manifolds can have a paring. On the other hand, as shown by the following classical result, an atoroidal three-manifold has a natural paring.

Proof. Since \(M\) is atoroidal, every Abelian, non cyclic subgroup of \(\pi_1(M)\) is conjugate to a subgroup of the fundamental group of a component of \(P\). It remains to check whether or not any incompressible cylinder \((C,\partial C)\subset (M,P)\) can be homotoped relatively to its boundary into \(P\). If \(P=\emptyset\) there is nothing to prove. Let us now assume that every torus component of \(\partial M\) is incompressible and that there is an incompressible cylinder \((C,\partial C)\subset (M,P)\) that can not be homotoped relative to its boundary into \(P\).

We have two cases to consider. In the first case, \(C\) joins two different components \(T_1\) and \(T_2\) of \(P\), then a regular neighborhood \({\mathcal{N}}\) of \(T_1\cup T_2\cup C\) is a product \(S^1\times S_{0,3}\) of a circle by a pair of pants. In the second case, the two boundary components of \(C\) lie in the same component \(T_1\) of \(T\) and a regular neighborhoud \({\mathcal{N}}\) of \(T_1\cup C\) is again a product \(S^1\times S_{0,3}\).

Let \(T_1, T_2\) and \(T_3\) be the components of \(\partial{\mathcal{N}}\). By construction \(T_1\) (and \(T_2\) in the first case) is a component of \(\partial M\). By the Loop theorem, if the inclusion \(T_i\to M\) is not \(\pi_1\)-injective then \(T_i\) bounds a solid torus for \(i\in\{1,2,3\}\). Since \(M\) is atoroidal, if \(T_i\) is incompressible for \(i=2\) or \(3\), then \(T_i\) can be homotoped into a boundary component \(T'_i\) of \(M\). Then, by [46], \(T_i\) and \(T'_i\) bound a thickened torus and \(T_i\) is isotopic to \(T_i'\). It follows that either \(M\) is homeomorphic to \({\mathcal{N}}\) or \(M\) is obtained by gluing a solid torus along \(T_3\) and/or \(T_2\). Since \(C\) is incompressible, the product structure of \({\mathcal{N}}\) extends to foliations by circles. Thus we have established that \(M\) is a Seifert fibered manifold, in particular \(M\) has a finite cover \(M'\) which is a circle bundle over a compact surface \(F\) (see [46]) such that \(M'\) and \(F\) are orientable. Since \(M\) is atoroidal, \(M'\) is also atoroidal, hence \(F\) has genus \(0\) and at most two boundary components (see [48]). Now \(M'\) is either a solid torus or a thickened torus. We have already ruled out the case of a solid torus so we are left with \(M\) being an \(I\)-bundle over a torus or Klein bottle. \(\rule{.55em}{.55em}\)

We say that \((M,P)\) is boundary irreducible if each component of \(\partial M\setminus P\) is incompressible and acylindrical if there is no essential disc or cylinder in \(M\) disjoint from \(P\).

2.2.2 Kleinian groups↩︎

A Kleinian group \(K\) is a discrete subgroup of \(\mathbb{P}SL_2(\mathbb{C})\) —the group of orientation preserving isometries of the \(3\)-dimensional hyperbolic space \(\mathbb{H}^3\). An orientable compact three-manifold \(M\) is hyperbolizable if its interior is homeomorphic to the quotient \(\mathbb{H}^3/K\) where \(K\) is a torsion free Kleinian group. Such a manifold \(M\) is irreducible and atoroidal and we say that \(M\) is uniformized by \(K\). Note that \(K\) is isomorphic to the fundamental group of \(M\), and that it is necessarily word hyperbolic if it contains no subgroup isomorphic to \(\mathbb{Z}\oplus\mathbb{Z}\), see Section 3.2. On the other hand, the tameness theorem [49], [50] asserts that, when \(K\) is a finitely generated torsion free Kleinian group, \(\mathbb{H}^3/K\) is homeomorphic to the interior of a compact three-manifold \(M_K\) (with fundamental group isomorphic to \(K\)) that we call the Kleinian manifold of \(K\).

As Poincaré observed, we may identify the Riemann sphere with the boundary at infinity of \(\mathbb{H}^3\) [51]. Then \(K\) acts on the Riemann sphere \(\widehat{\mathbb{C}}\) via Möbius transformations. The latter action is usually not properly discontinuous: there is a canonical and invariant partition \[\widehat{\mathbb{C}}= \Omega_K \sqcup \Lambda_K\] where \(\Omega_K\) denotes the domain of discontinuity, also called ordinary set, which is the largest open set of \(\widehat{\mathbb{C}}\) on which \(K\) acts properly discontinuously, and where \(\Lambda_K\) denotes the limit set, which is the set of accumulation points in \(\widehat{\mathbb{C}}\) of an (any) orbit \(Kp\subset\mathbb{H}^3\). When \(\Lambda_K\) has at least three points, implying that it is infinite (we say that \(K\) is non-elementary), it is also the minimal non-empty \(K\)-invariant compact subset of \(\widehat{\mathbb{C}}\). The construction of the Kleinian manifold \(M_K\) induces an embedding \((\mathbb{H}^3\cup \Omega_K)/K\hookrightarrow M_K\) whose image is the complement of a subsurface of \(\partial M_K\).

If \(\Lambda_K\) contains at least \(2\) points, the group \(K\) preserves the convex hull \(Hull(\Lambda_K)\) of its limit set in \(\mathbb{H}^3\). The group \(K\) is convex-cocompact if its action is cocompact on \(Hull(\Lambda_K)\) and \(K\) is geometrically finite if a regular neighborhood of a fundamental domain for the action of \(K\) on \(\mathrm{Hull}(\Lambda_K)\) has finite volume or if the limit set consists of a singleton. Ahlfors [52] showed that if the limit set of a geometrically finite Kleinian group is not the whole Riemann sphere then it has measure 0 (this holds more generally for finitely generated Kleinian groups by [53], [49] and [50]). When \(K\) is not Fuchsian, i.e., \(Hull(\Lambda_K)\) is not contained in a geodesic plane, and \(K\) is torsion free, there is a homeomorphism between \((\mathbb{H}^3\cup \Omega_K)/K\) and the convex core \(Hull(\Lambda_K)/K\) constructed using the closest point projection. As we have explained above \((\mathbb{H}^3\cup \Omega_K)/K\) embeds in \(M_K\), hence the convex core also embeds in the Kleinian manifold. When \(K\) is geometrically finite, the image of this embedding is the complement of a paring \(P\) of \(M_K\) corresponding to the parabolics of \(K\) (notice that this gives an alternate characterization of geometric finiteness). We say that a torsion free geometrically finite Kleinian group \(K\) uniformizes the pared manifold \((M,P)\) when \((\mathbb{H}^3\cup \Omega_K)/K\) is homeomorphic to \(M\setminus P\).

Now it follows from the Loop theorem (see [46] for example) that \(\partial M\setminus P\) is incompressible if and only if each connected component of \(\Omega_K\) is simply connected. This leads to:

Thurston’s uniformization theorem (extended to non Haken manifolds by Perel’man) gives a topological characterization of hyperbolic manifolds. We will use the following form, see [54], and also [48], [55], [56]:

When \((M,P)\) is acylindrical, then a doubling argument shows that we can require the convex core to have totally geodesic boundary [57].

::: {#thm:thurston acylindrical .theorem} Theorem 15. Let \((M,P)\) be a Haken acylindrical pared three-manifold. There is a geometrically finite, complete hyperbolic manifold \(N\) whose convex core is homeomorphic to \(M\setminus P\) and has totally geodesic boundary. :::

We say that a Kleinian group is minimally parabolic if every parabolic subgroup is a rank \(2\) Abelian subgroup. Combining Theorem 14 with Scott’s core theorem we get that any finitely generated Kleinian group has a minimally parabolic version:

Proof. Let \(K\) be a torsion-free finitely generated Kleinian group. By Scott’s core theorem [58], the hyperbolic manifold \(\mathbb{H}^3/K\) contains a compact three-dimensional submanifold \(C\) such that the inclusion is a homotopy equivalence. If \(\partial C=\emptyset\), then \(C=\mathbb{H}^3/K\) and \(\mathbb{H}^3/K\) is compact. Thus, \(K\) has no parabolic subgroup and there is nothing to prove. So let us assume that \(\partial C\neq\emptyset\), in particular, \(C\) is Haken. A maximal Abelian non cyclic subgroup of \(\pi_1(C)\) corresponds to a rank \(2\) parabolic subgroup of \(K\). By [59] we can change \(C\) by a homotopy so that such a subgroup is the fundamental group of a component of \(\partial C\). It follows that \(C\) is atoroidal. Now if \(P\) is the union of the tori in \(\partial C\), then \((C,P)\) is a pared manifold and we conclude with Theorem 14. \(\rule{.55em}{.55em}\)

3 Hyperbolicity in the sense of Gromov↩︎

Hyperbolic spaces and groups were introduced by Gromov in [60] and have known many developments since. In this section, we will introduce hyperbolic spaces, hyperbolic and relatively hyperbolic groups and a few more objects associated to these groups. We will also establish various facts that will be used throughout the paper. We start with hyperbolic spaces and word hyperbolic groups. Background on those includes [1], [60], [61].

3.1 Hyperbolic spaces↩︎

Let \(X\) be a metric space. It is geodesic if any pair of points \(\{x,y\}\) can be joined by a (geodesic) segment i.e, a map \(\gamma:[0,d(x,y)]\to X\) such that \(\gamma(0)=x\), \(\gamma(d(x,y))=y\) and \(d(\gamma(s),\gamma(t))=|t-s|\) for all \(s,t\in [0,d(x,y)]\). The metric space \(X\) is proper if closed balls of finite radius are compact.

A triangle \(\Delta\) in a metric space \(X\) is given by three points \(\{x,y,z\}\) and three (geodesic) segments (or sides) joining them. Given a constant \(\delta\ge 0\), the triangle \(\Delta\) is \(\delta\)-slim if any side of the triangle is contained in the \(\delta\)-neighborhood of the two others.

Basic examples of hyperbolic spaces are the complete simply connected hyperbolic manifolds \(\mathbb{H}^n\), \(\mathbb{R}\)-trees and their convex subsets. It follows from the shadowing lemma (see below) that, among geodesic metric spaces, hyperbolicity is invariant under quasi-isometry : if \(X\), \(Y\) are two quasi-isometric geodesic metric spaces, then \(X\) is hyperbolic if and only if \(Y\) is hyperbolic. A \((\lambda,c)\)-quasigeodesic is the image of an interval (that can be unbounded on both sides) by a \((\lambda,c)\)-quasi-isometric embedding.

3.1.1 Boundaries of hyperbolic spaces↩︎

A proper (geodesic) hyperbolic space \(X\) admits a canonical compactification \(X\sqcup \partial X\) at infinity. This compactification can be defined by looking at the set of geodesic rays, i.e., isometric embeddings \(r:\mathbb{R}_+\to X\), up to bounded Hausdorff distance. The topology is induced by the uniform convergence on compact subsets of \(\mathbb{R}_+\): fix a base point \(w\in X\); for each point \(x\in X\), choose arbitrarily a geodesic segment \(s_x:[0,d(w,x)]\to X\) between \(w\) and \(x\) so that \(s_x(0)=w\) and extend it to \(r_x:\mathbb{R}_+\to X\) by setting \(r_x(t)=x\) if \(t\ge d(w,x)\). A sequence \((x_n)_n\) tends to a point at infinity if \(d(w,x_n)\longrightarrow\infty\) and all the limits of converging (uniformly on compact sets) subsequences of \((r_{x_n})_n\) have their images at bounded Hausdorff distance to each other. This is independent from the choice of the base point. The boundary can be endowed with a family of visual distances \(d_v\) compatible with its topology, i.e., for any base point \(w\in X\), there exists a constant \(C>0\) which satisfies \[(1/C) e^{-\varepsilon d(w , (a,b))} \le d_v(a,b) \le C e^{-\varepsilon d(w , (a,b))}\] where \(\varepsilon>0\) is a visual parameter chosen small enough, and \((a,b)\) is any geodesic asymptotic to rays defining \(a\) and \(b\).

If \(\Phi:X\to Y\) is a quasi-isometry between two proper hyperbolic spaces, then the shadowing lemma implies that \(\Phi\) induces a homeomorphism \(\phi:\partial X\to\partial Y\). Actually, the trace map at infinity of a quasi-isometry is quasi-Möbius [62], i.e., there exists a homeomorphism \(\theta:\mathbb{R}_+\to\mathbb{R}_+\) such that, for any distinct points \(x_1,x_2,x_3,x_4\in \partial X\), \[[\phi(x_1):\phi(x_2):\phi(x_3):\phi(x_4)]\le \theta([x_1:x_2:x_3:x_4])\] where \[[x_1:x_2:x_3:x_4]= \frac{d_v(x_1,x_2)\cdot d_v(x_3,x_4)}{d_v(x_1,x_3)\cdot d_v(x_2,x_4)}\,.\] Quasi-Möbius maps are stable under composition.

Quasi-isometries provide natural examples of quasi-Möbius maps, cf. [63]:

3.1.2 Groups of isometries and convergence actions↩︎

Let us consider a group \(G\) acting by homeomorphisms on a compact metrisable space \(Z\) with at least \(3\) points. We say that this action is a convergence group action if its diagonal action on the set of distinct triples is properly discontinuous [65]. The limit set \(\Lambda_G\) is by definition the set of accumulation points of an (any) infinite orbit \(Gp\subset Z\). One may consult [66] for basic properties of convergence groups.

Let us now consider a group \(G\) of isometries of a proper hyperbolic space \(X\). It follows from Theorem 19 that the action of \(G\) extends to an action on \(\partial X\) by uniform quasi-Möbius mappings.

If the action of \(G\) is furthermore properly discontinuous on \(X\), then the action of \(G\) on \(X\cup \partial X\) and on \(\partial X\) is a convergence action [67], [68].

3.2 Word hyperbolic groups↩︎

A properly discontinuous group action by isometries on a proper geodesic metric space is geometric if it is cocompact, i.e., if the quotient is compact.

We note that since the hyperbolicity of a proper geodesic hyperbolic metric space is a quasi-isometric invariant property, the hyperbolicity of a group does not depend on the space it is acting upon. In particular, by the Švarc–Milnor lemma, \(G\) is finitely generated and its hyperbolicity can also be read from any of its locally finite Cayley graphs. Moreover, this implies the quasi-isometric rigidity of the class of word hyperbolic groups.

Fundamental groups of closed hyperbolic manifolds are obvious examples. Convex cocompact Kleinian groups are also hyperbolic by definition: a convex-compact Kleinian group \(G\) has a properly discontinuous and cocompact action on the convex hull \(Hull(\Lambda_G)\) of its limit set which is a hyperbolic space in the sense of Gromov.

The definition and the Švarc–Milnor lemma also imply that a word hyperbolic group \(G\) admits a topological boundary \(\partial G\) defined by considering the boundary of any proper geodesic metric space on which \(G\) acts geometrically. In the case of a convex-cocompact Kleinian group \(K\), a model for the boundary \(\partial K\) is given by its limit set \(\Lambda_K\).

The action of a hyperbolic group on its boundary is a uniform convergence action, i.e., its diagonal action on the set of distinct triples is not only properly discontinuous but also cocompact, cf. [66].

These properties characterize word hyperbolic groups and their boundaries:

A general principle asserts that a word hyperbolic group is determined by its boundary. More precisely, Paulin proved that the quasi-isometry class of a word hyperbolic group is determined by its boundary equipped with its quasiconformal structure [63].

3.3 Relative hyperbolicity↩︎

The idea behind relatively hyperbolic groups which will be defined next is that some metric spaces such as Cayley graphs are hyperbolic away from some “codimension 1” subspaces or subgroups. Background on relatively hyperbolic groups includes [60], [70], [71] and the references therein. Let us first recall the definition of the Busemann function \(\beta_p(x,y)\) centered at a point \(p\in \partial X\) at infinity of a hyperbolic space \(X\) measured at two points \(x,y\in X\): \[\beta_p(x,y)=\sup \left\{\lim_{t\to\infty} [ d(x,\gamma(t)) -t]:\gamma geodesic ray asymptotic to psuch that \gamma(0)=y\right\}\,.\]

Given \(p\in \partial X\), a base point \(w\in X\) and \(r\in\mathbb{R}\), the horoball centered at \(p\) of level \(r\) is defined as \[H_w(p,r)=\{x\in X: {\beta_p(x,w)}\le r\}\,.\]

Let \(G\) be a group and let \(\mathbb{P}\) be a collection of subgroups. The pair \((G,\mathbb{P})\) is relatively hyperbolic if there exists a hyperbolic proper geodesic metric space \(X\) on which \(G\) acts properly discontinuously by isometries and if there is a \(G\)-invariant collection \({\mathcal{H}}\) of pairwise disjoint horoballs with the following properties:

1. any \(P\in\mathbb{P}\) is the stabilizer of the center (at infinity) of a horoball in \({\mathcal{H}}\);

2. the stabilizer of any center of a horoball of \({\mathcal{H}}\) is conjugate to a subgroup from \(\mathbb{P}\);

3. the action of \(G\) on \(X\setminus Y\) is cocompact, where we let \(Y\) denote the union of the horoballs in \({\mathcal{H}}\).

Such an action of \(G\) is called cusp uniform. Note that we may assume that no two elements of \(\mathbb{P}\) are conjugate, so that \(\mathbb{P}\) must be finite. The subgroups in \(\mathbb{P}\) are called peripheral subgroups. Note also that, in this case, the limit set of the action is the whole boundary \(\partial X\).

3.3.1 Horoballing, cusped spaces and Bowditch boundaries↩︎

In this section, we introduce an explicit cusp uniform action for a relatively hyperbolic group. We first concentrate on the construction of horoballs resting on peripheral subgroups. Let \((P,d)\) be a graph endowed with the path metric that makes each edge isometric to \([0,1]\). Define its horoballing space \(H_P\) following Groves and Manning [72] as the graph modelled on \(P\times \mathbb{N}\) with additional edges \[\begin{pmatrix} x\\ m\end{pmatrix} \sim \begin{pmatrix} y\\ m\end{pmatrix}\quadif \quad d(x,y)\le 2^m\] and \[\begin{pmatrix} x\\ m\end{pmatrix}\sim \begin{pmatrix} x\\ m+1\end{pmatrix}\] This new graph \(H_P\) is also endowed with the path metric so that each edge is isometric to \([0,1]\). By [72], \(H_P\) is hyperbolic.

Let \(G\) be a finitely generated group, let \(\mathbb{P}=\{P_1,...,P_n\}\) be a (finite) family of finitely generated subgroups of \(G\), and let \(S\) be a finite generating set for \(G\) so that \(P_i\cap S\) generates \(P_i\) for each \(i\in\{1,...,n\}\) , and denote by \(\mathcal{C}\!\it ay\,(G,S)\) the Cayley graph of \((G,S)\). For each \(i\in\{1,...,n\}\), let \(T_i\) be a left transversal for \(P_i\), i.e., a collection of representatives for left cosets of \(P_i\) in \(G\) which contains exactly one element of each left coset. For each \(i\), and each \(t\in T_i\), let \(\mathcal{C}\!\it ay\,_{i,t}\) be the full subgraph of the Cayley graph \(\mathcal{C}\!\it ay\,(G,S)\) which contains \(tP_i\). Each \(\mathcal{C}\!\it ay\,_{i,t}\) is isomorphic to the Cayley graph of \(P_i\) with respect to the generators \(P_i\cap S\). We define the cusped space \[\mathcal{C}\!\it us\,(G,\mathbb{P},S)=\mathcal{C}\!\it ay\,(G,S)\cup(\{\cup H_{\mathcal{C}\!\it ay\,_{i,t}}|1\leq i\leq n, t\in T_i\}),\] where the graphs \(\mathcal{C}\!\it ay\,_{i,t}\subset \mathcal{C}\!\it ay\,(G,S)\) and \(\mathcal{C}\!\it ay\,_{i,t}=\mathcal{C}\!\it ay\,_{i,t}\times\{0\}\subset H_{\mathcal{C}\!\it ay\,_{i,t}}\) are identified in the obvious way. When the choice of the generating set does not matter, we simplify the notations to \(\mathcal{C}\!\it ay\,(G)\) and \(\mathcal{C}\!\it us\,(G,\mathbb{P})\).

The cusped space provides a canonical way to construct a hyperbolic space on which a relatively hyperbolic group has a cusp uniform action. We follow Groves and Manning [72] in using this space to give a different characterization of relatively hyperbolic groups.

The horoballings \(H_{\mathcal{C}\!\it ay\,_{i,t}} \subset \mathcal{C}\!\it us\,(G,\mathbb{P},S)\), \(t\in G\), \(P_i\in P\) are horoballs and the action of \(G\) on \(\mathcal{C}\!\it us\,(G,\mathbb{P},S)\) is cusp uniform. The cusped space provides us a way to define a boundary for relatively hyperbolic groups.

Corollary 1 below shows that the boundary is well defined, as a metric space, up to a quasi-Möbius change of metrics. Note that the topology of the boundary is well-defined according to [70]. Reference to a generating set is thus unnecessary.

In the next sections we will discuss quasi-isometries between relatively hyperbolic groups. In particular we will show that the previous definition does not depend on the choice of the generating set \(S\).

3.3.2 Geometrically finite actions↩︎

Let \(G\) be a convergence group acting on some metrizable compact space \(Z\). An element \(g\in G\) is loxodromic if it has infinite order and fixes exactly two points of \(Z\). A subgroup \(H<G\) is parabolic if it is infinite and has a unique fixed point. We refer to it as a parabolic point. A parabolic subgroup contains no loxodromics [66]. The stabilizer of a parabolic point is necessarily a parabolic group. There is thus a natural bijective correspondence between parabolic points in \(Z\) and maximal parabolic subgroups of \(G\). We say that a parabolic group, \(H\), with fixed point \(p\), is bounded if the quotient \((\Lambda_G \setminus \{p\})/H\) is compact. (It is necessarily Hausdorff.) We say that \(p\) is a bounded parabolic point if its stabilizer is bounded. A conical limit point is a point \(y\in Z\) such that there exists a sequence \((g_j)_{j\ge 0}\) in \(G\), and distinct points \(a, b\in Z\), such that \(g_j(y)\) tends to \(a\) and \(g_j(x)\) tends to \(b\) for all \(x \in Z\setminus \{y\}\). We finally say that the action of \(G\) on \(Z\) is geometrically finite if every point of its limit set \(\Lambda_G\) is either conical or bounded parabolic (they cannot be both simultaneously). When \(G\) is a Kleinian group and \(Z=S^2\), this definition is equivalent to the one given in §2.2.2, see for instance [73] and the references therein.

When a relatively hyperbolic group \((G,\mathbb{P})\) admits a cusp uniform action on a proper geodesic hyperbolic space \(X\) then its action on the boundary \(\partial X\) is geometrically finite, cf. [70]. The conjugates of subgroups in \(\mathbb{P}\) are precisely the maximal parabolic subgroups. Conversely, if a group \(G\) admits a geometrically finite action on a metrizable space \(Z\), then the pair \((G,\mathbb{P})\) is relatively hyperbolic [74], where \(\mathbb{P}\) denotes a set of conjugacy representatives of maximal parabolic subgroups of \(G\). It turns out that the topology of \(\Lambda_G\) is independent of the space \(Z\) as long as the action is geometrically finite with the same parabolic subgroups [70]. See [71] for more information.

A subgroup \(H\) of a relatively hyperbolic group \((G,\mathbb{P})\) is elementary if, whenever \((G,\mathbb{P})\) admits a cusp uniform action on a proper geodesic hyperbolic space, the limit set of \(H\) has at most two points.

3.4 Horoballs↩︎

Let us set some facts about the metrics on the combinatorial horoballs \(H_P\) defined above.

Proof. By construction \(H_P\) is a geodesic space. We denote by \(\ell(.)\) the length of a path, i.e., the number of edges it contains. A horizontal path will denote a path contained in some level \(P\times\{ n\}\) for some \(n\ge 0\) and a vertical path will be of the form \(\{p\}\times [m,n]\) for some \(p\in P\) and some integers \(n\ge m\ge 0\). Denote by \(\left[\begin{pmatrix} x\\ n\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right]_h\) a horizontal path with minimal length and by \(\left[\begin{pmatrix} x\\ m\end{pmatrix},\begin{pmatrix} x\\ n\end{pmatrix}\right]_v\) a vertical path.

It is easy to see that \[\label{eq:x61y} d\left[\begin{pmatrix} x\\ n\end{pmatrix},\begin{pmatrix} x\\ m\end{pmatrix}\right]=|m-n|\tag{1}\]

When \(x\neq y\), we remark that: \[\ell\left(\left[\begin{pmatrix} x\\ n\end{pmatrix},\begin{pmatrix} x\\ n+1\end{pmatrix}\right]_v\cup \left[\begin{pmatrix} x\\ n+1\end{pmatrix},\begin{pmatrix} y\\ n+1\end{pmatrix}\right]_h\right)\leq \ell\left(\left[\begin{pmatrix} x\\ n\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right]_h\cup \left[\begin{pmatrix} y\\ n\end{pmatrix},\begin{pmatrix} y\\ n+1\end{pmatrix}\right]_v\right)\] Therefore, if \(\begin{pmatrix} x\\ m\end{pmatrix}\) and \(\begin{pmatrix} y\\ n\end{pmatrix}\) are two points in \(H_P\), then they are joined by a geodesic of the form \[\left[\begin{pmatrix} x\\ m\end{pmatrix},\begin{pmatrix} x\\ k\end{pmatrix}\right]_v\cup \left[\begin{pmatrix} x\\ k\end{pmatrix},\begin{pmatrix} y\\ k\end{pmatrix}\right]_h\cup\left[\begin{pmatrix} y\\ k\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right]_v\] with \(k\ge \max\{m,n\}\). Hence \[d\left[\begin{pmatrix} x\\m\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right]=\min_{k\geq\max\{m,n\}}\{2k-m-n+\lceil2^{-k}{d_P}(x,y)\rceil\}\] with \(\lceil a\rceil =p\) if \(p-1< a\leq p\) for some \(p\in\mathbb{Z}\).

Set \(j=\lfloor\log_2({d_P}(x,y))\rfloor-k\) then \[2k-m-n+\lceil 2^{-k}{d_P}(x,y)\rceil=2\lfloor\log_2({d_P}(x,y))\rfloor-m-n-2j+\lceil 2^{j+\varepsilon(x,y)}\rceil\] where \(\varepsilon(x,y)=\log_2({d_P}(x,y))-\lfloor\log_2({d_P}(x,y))\rfloor\).

Set \(f_\varepsilon(j)=-2j+\lceil 2^{j+\varepsilon}\rceil\). \[\label{value32min} \text{For } 0\leq\varepsilon<1,\text{let } g(\varepsilon)=\min\{f_\varepsilon(j),j\in\mathbb{Z}\}\in\{0,1,2\}\tag{2}\] This minimum is always reached for \(j=1\) (it might also be reached for \(j=0\) or \(j=2\) depending on the value of \(\varepsilon\)) and \(f_\varepsilon\) is decreasing for \(j\leq 1\). Set \(A=\lfloor\log_2({d_P}(x,y))\rfloor-\max\{m,n\}\). It follows that \[\label{eq:k32petit} d\left[\begin{pmatrix} x\\m\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right]=|m-n|+ \lceil 2^{A+\varepsilon(x,y)}\rceil \text{ when } {A\leq 1}\tag{3}\] and \[\label{eq:k32grand} d\left[\begin{pmatrix} x\\ m\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right]=2\lfloor\log_2({d_P}(x,y))\rfloor-(m+n)+g(\varepsilon(x,y)) \text{ when } A\geq 1.\tag{4}\]

We conclude the proof by combining equations (1 ), (3 ), (4 ) and (2 ) with the fact that \(\log_2 a\leq \log_2 (1+a)\leq \max\{1,\log_2 a\} + 1\) for \(a >0\).

\(\rule{.55em}{.55em}\)

We recall the following fact that we leave as an exercise for the reader.

From this fact, we will deduce:

Fact 28 is also used to extend quasi-isometric embeddings between metric spaces to their horoballings, see also [75].

When needed, we will apply Fact 28 to subgroups as follows. Given a finitely generated subgroup \(H\) of a finitely generated group \(G\), consider a finite generating set \(S\) for \(G\) so that \(S\cap H\) generates \(H\). Let \(X\) be the Cayley graph of \((G,S)\) and \(Y\) the full subgraph with vertex set \(H\).

The subspace \(Y\subset X\) is path connected and we may construct a function \(h\) satisfying the assumption of Fact 28.

Proof. We assume that \(f:X\to X'\) and its quasi-inverse \(g:X'\to X\) are \((\lambda,c)\)-quasi-isometries and that \(d(g\circ f(x),x)\le c\) for all \(x\in X\). Let us denote the Hausdorff distance by \(d_H\) and let us suppose that \(d_H(f(Y),Y')\le D\).

For any \(x\in Y\), there exists \(\varphi(x)\in Y'\) such that \(d(f(x),\varphi(x))\le D\). This defines a map \(\varphi:Y\to Y'\). Given \(x,y\in Y\) let \(k\subset Y\) be an arc joining \(x\) to \(y\) such that \(\ell(k)\leq \lfloor d_Y(x,y)\rfloor+1\). On \(k\), we consider \(p+1\) points \(x=x_0, x_1,\ldots, x_p=y\) such that \(p=\lfloor d_Y(x,y)\rfloor +1\) and \(d_Y(x_i,x_{i+1})\leq 1\). By construction, we have \(d(\varphi(x_i),\varphi(x_{i+1}))\leq 2D+\lambda+c\). It follows that \(d_{Y'}(\varphi(x_i),\varphi(x_{i+1}))\leq\sup_{t\leq 2D+\lambda+c} h(t)=K\) and we get \[d_{Y'}(\varphi(x),\varphi(y))\leq p K\leq Kd_Y(x,y)+K.\]

Moreover, \[\begin{align} d_H(g(Y'),Y) & \le & d_H(g(Y'), g(f(Y)) ) + d_H((g\circ f)(Y), Y) \\ & \le & (\lambda d_H(Y',f(Y)) + c )+ c\\ &\le & \lambda D +2c \end{align}\] Therefore, letting \(\psi:Y'\to Y\) satisfy \(d(\psi(x'),g(x'))\le \lambda D +2c\) for all \(x'\in Y'\), we obtain as above \(K'\) such that: \[d_{Y'}(\psi(x'),\psi(y')) \le K' d(x',y') + K'\] Additionally, for any \(x\in Y\) \[\begin{align} d(\psi\varphi(x),x) & \le & d(\psi\varphi(x), g\varphi(x) ) + d(g\varphi(x), gf(x)) + d(gf(x),x) \\ & \le & (\lambda D +2c) + (\lambda D + c ) + c\\ &\le & 2\lambda D +{4}c \end{align}\] We prove similarly that \(d(\varphi\psi,id)\) is bounded so that Fact 27 applies.\(\rule{.55em}{.55em}\)

Proof. Let \(\lambda,c\) be such that \(\forall x,y\in P\), \(\lambda^{-1} d(f(x),f(y))-c\leq d(x,y)\leq \lambda d(f(x),f(y))+c\).

Then we have: \[\begin{align} d\left[F\begin{pmatrix} x\\ m\end{pmatrix},F\begin{pmatrix} y\\ n\end{pmatrix}\right] & \le & \max\{ |m-n| , 2 \log_2(1+ \lambda d(x,y)+c) - (m+n) \}+ O(1)\\ & \le & \max\left\{ |m-n|, 2\log_2 \left(\frac{1+c}{\lambda}+ d(x,y)\right) - (m+n) \right\} + O(1) \end{align}\] If \(1+c\le\lambda\), we have \(\log_2 \left(\displaystyle\frac{1+c}{\lambda}+ d(x,y)\right)\leq \log_2 (1+ d(x,y))\). Otherwise, \[\begin{align} \log_2 \left(\frac{1+c}{\lambda}+ d(x,y)\right) & = & \log_2\left(\frac{1+c}{\lambda}\right)+\log_2\left(1+\frac{\lambda}{1+c}d(x,y)\right)\\ & \leq & \log_2(1+d(x,y))+\log_2\left(\frac{1+c}{\lambda}\right)\,. \end{align}\] In both cases, we get

\[\begin{align} d\left[F\begin{pmatrix} x\\ m\end{pmatrix},F\begin{pmatrix} y\\ n\end{pmatrix}\right] & \le & d\left[\begin{pmatrix} x\\ m\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right] +O(1)\,. \end{align}\]

On the other hand, if \(1+\lambda^{-1} d(x,y)-c>{1}\), we have \[\begin{align} d\left[F\begin{pmatrix} x\\ m\end{pmatrix},F\begin{pmatrix} y\\ n\end{pmatrix}\right] & \ge & \max\{ |m-n| , 2 \log_2(1+ \lambda^{-1} d(x,y)-c) - (m+n) \}+ O(1)\\ &\ge & d\left[\begin{pmatrix} x\\ m\end{pmatrix},\begin{pmatrix} y\\ n\end{pmatrix}\right] +O(1)\,, \end{align}\] where we have used that \(\log_2(1+ \lambda^{-1} d(x,y)-c) \ge \log_2 (1+ d(x,y)) + \log_2 1/(1+ c\lambda)\) since \(d(x,y)\ge c\lambda\). If \(1+\lambda^{-1} d(x,y)-c\le 1\), then both \(d(x,y)\) and \(d(f(x),f(y))\) are bounded, so the result follows. \(\rule{.55em}{.55em}\)

The following fact is then easily deduced from Facts 27 and 29:

We now prove an analogous statement for horoballs in real hyperbolic space.

Proof. Note that in the upper half-space model, \[\begin{align} d_{\mathbb{H}^{d+1}}((z,y),(z',y'))& =& \rm arcosh \left(1 + \frac{\|z-z'\|_E^2+(y-y')^2}{2yy'}\right)\\ &=&\ln\left(1 + \frac{\|z-z'\|_E^2+(y-y')^2}{2yy'}\right) + O(1) \end{align}\]

For any \(a,b> 0\), \(\max\{\ln a,\ln b\}\leq \ln(a+b)\leq \max\{\ln(2a),\ln(2b)\}\), hence \(\ln(a+b)=\max\{\ln a,\ln b\}+O(1)\). Applied to the previous equality, this gives

\[\begin{align} d_{\mathbb{H}^{d+1}}((z,y),(z',y'))& = & \max\left\{ \ln \left(\frac{\|z-z'\|_E^2}{2yy'}\right) , \ln\left(1 + \frac{(y-y')^2}{2yy'}\right) \right\}+ O(1)\\ & = & \max\left\{ 2\ln (\|z-z'\|_E) -\ln {(yy')} , \ln\left(\frac{y}{2y'}+\frac{y'}{2y} \right) \right\}+ O(1) \end{align}\]

Let us assume that \(y,y'\geq 1\), so that \(\ln(yy')\geq 0\). When \(\|z-z'\|_E\leq 1\), the term \(2\ln (\|z-\nolinebreak z'\|_E) -\ln {(yy')}\) above is irrelevant. From the fact that, for any \(a\geq 1\), \(\ln a=\ln(1+a)+O(1)\), we deduce

\[\begin{align} d_{\mathbb{H}^{d+1}}((z,y),(z',y'))& = & \max\left\{ 2\ln (1 +\|z-z'\|_E) -\ln {yy'} , \left|\ln \left(\frac{y}{y'}\right)\right| \right\}+ O(1) \,. \end{align}\]

Therefore, since \(\{y_n\}\) is bounded, \[\begin{align} d_{\mathbb{H}^{d+1}}((z,y_n 2^n),(z',y_m 2^m)) & = & \max\left\{ 2\ln (1 +\|z-z'\|_E) - (m+n)\ln 2) , |m-n| \ln 2 \right\}+O(1)\\ & = & \ln 2 \max\left\{ 2\log_2 (1 +\|z-z'\|_E) - (m+n)) , |m-n| \right\} + O(1)\,. \end{align}\] The rest of the proof follows as for Fact 29. \(\rule{.55em}{.55em}\)

3.4.1 Pared groups↩︎

We adapt the notion of pared three-manifolds to groups. This point of view was first developed by Otal for free groups in [76].

Let \(G\) be a finitely generated group. A paring will be given by a finite almost malnormal collection of subgroups \(\mathbb{P}_G=\{P_1,\ldots, P_k\}\); almost malnormal means that if \(P_i\cap (gP_jg^{-1})\) is infinite for some \(i,j\in\{1,\ldots, k\}\) and \(g\in G\), then \(i=j\) and \(g\in P_i\). For example if \((G,\mathbb{P}_G)\) is relatively hyperbolic, then it is a pared group. A pared compact three-manifold \((M,P_M)\) gives rise to a canonical pared group \((K,\mathbb{P}_K)\) by letting \(K=\pi_1(M)\) and \(\mathbb{P}_K\) denote a representative for each conjugacy class of subgroups corresponding to the fundamental groups of the annuli and tori composing \(P_M\). Let us call this pared group the pared fundamental group of \((M,P_M)\).

To mimic more general \(3\)-manifolds with boundary we want to loosen the condition on \(\mathbb{P}_G\). An adornment is a finite collection of subgroups \(\mathbb{P}_G=\{P_1,\ldots, P_k\}\) that are non coarsely nested, i.e., given \(P_i,P_j\in \mathbb{P}_G\) and \(g\in G\), if \(g_iP_i\) is contained in some \(D\)-neighborhood of \(P_j\) for some \(D<\infty\) then \(P_i=P_j\) and \(g \in P_j\). It is not difficult to prove that a pared group is also adorned. A Seifert manifold \(M\) with boundary admitting an \(\mathbb{H}^2\times \mathbb{E}^1\) structure gives rise to a canonical adorned group \((\pi_1(M),\mathbb{P}_M)\) by letting \(\mathbb{P}_M\) denote the collection of the fundamental groups of the components of \(\partial M\). The adorned fundamental group \((\pi_1(M),\mathbb{P}_M)\) is not a pared group since \(M\) is Seifert fibered and the fibers induce a normal cyclic subgroup that is a subgroup of all elements in \(\mathbb{P}_M\).

An isomorphism between group pairs \((G,\mathbb{P}_G)\) and \((Q,\mathbb{P}_Q)\) is an isomorphisms \(G\to Q\) that maps the conjugacy classes in \(\mathbb{P}_G\) into the conjugacy classes in \(\mathbb{P}_Q\).

A quasi-isometry between two adorned groups \((G,\mathbb{P}_G)\) and \((H,\mathbb{P}_H)\) will be given by a quasi-isometry \(\Phi:G\to H\) such for any \(g\in G\) and \(P\in\mathbb{P}_G\) there is \(h\in H\) and \(Q\in \mathbb{P}_H\) such that \(\Phi(gP)\) is at bounded Hausdorff distance from \(hQ\) and for any \(h\in H\) and \(Q\in \mathbb{P}_H\) there is \(g\in G\) and \(P\in\mathbb{P}_G\) such that \(\Phi(gP)\) is at bounded Hausdorff distance from \(hQ\). The non coarse nestedness of the subgroups guarantees the uniqueness of \(h\) and \(Q\) in the first part and of \(g\) and \(P\) in the second. Note that if \(\mathbb{P}_G\) is a paring for \(G\), then the quasi-isometry implies that \(\mathbb{P}_H\) is also almost malnormal. We will say that a pared manifold \((M,P_M)\) is quasi-isometric to a pared group \((G,\mathbb{P}_G)\) if there is a quasi-isometry between its pared fundamental group \((K,\mathbb{P}_K)\) and \((G,\mathbb{P}_G)\).

In the sequel, it will be convenient to write that two subsets are at bounded distance to mean they are at bounded Hausdorff distance.

Observe that since \(H\) has finite index, each transversal \(T_P\) is finite.

Let us check that \((H,\mathbb{Q})\) is an adorned group. Take \(h\in H\), \(a_i\in T_{P_i}\), \(a_j\in T_{P_j}\) and \(P_i,P_j\in\mathbb{P}\) and assume that \(a_iP_ia_i^{-1}\cap H\) is contained in a finite neighborhood of \(h(a_jP_ja_j^{-1}\cap H)\). Since \(H\) has finite index, it follows that \(a_iP_ia_i^{-1}\) is contained in a finite neighborhood of \(h(a_jP_ja_j^{-1})\). Moreover, \(a_iP_ia_i^{-1}\) is at bounded distance from \(a_iP_i\), and similarly for \(j\) so that \(a_iP_i\) is contained in a finite neighborhood of \(ha_jP_j\). Therefore, \(P_i=P_j\) and \(ha_j \in a_i P_i\), so \(a_j\in Ha_i P_i\) implying that \(a_i=a_j\) and \(h\in a_iP_ia_i^{-1}\cap H\). This means that \((H,\mathbb{Q})\) is an adorned group.

If \(HaP=HbP\) for some \(P\in\mathbb{P}\) and \(a,b\in G\), then there exists \(h\in H\) and \(p\in P\) such that \(b= hap\). Therefore \[bPb^{-1}\cap H = hapPp^{-1}a^{-1}h^{-1}\cap H = h(aPa^{-1}\cap H) h^{-1}\] thus \(bPb^{-1}\cap H\) is conjugate in \(H\) to \(aPa^{-1}\cap H\). This means that the choice made when picking \(T_P\) only determines the representatives of the conjugacy classes of the elements in \(\mathbb{Q}\).

Proof. Let us first prove that the canonical injection defines a quasi-isometry between adorned groups. Let us fix a word metric \(|\cdot |\) on \(G\) provided by a finite generating set. We shall denote by \(d(\cdot,\cdot)\) the Hausdorff distance between non-empty subsets of \(G\).

Let us observe that as \(\mathbb{P}\) and \(T_P\) are finite sets and each subgroup \(aPa^{-1}\cap H\) has finite index in \(aPa^{-1}\) for all \(P\in\mathbb{P}\) and \(a\in T_P\), there is a constant \(B\) such that \(d(aPa^{-1}\cap H,aPa^{-1})\le B\) for all \(P\in\mathbb{P}\) and \(a\in T_P\). We may as well assume that \(|a|\le B\) holds for every \(a\in T_P\). Thus, by the triangle inequality, we have \[d(aPa^{-1}\cap H, aP) \le d(aPa^{-1}\cap H, aPa^{-1}) + d(aPa^{-1},aP)\le B +|a| \le 2B\,.\]

Let us start by showing that each coset \(h(aPa^{-1}\cap H)\) in \(H\), where \(P\in \mathbb{P}\), \(a\in T_P\) and \(h\in H\), is at bounded distance from the coset \((ha)P\) in \(G\). We have \[d(h(aPa^{-1}\cap H), (ha)P)= d(aPa^{-1}\cap H, aP) \le 2B\,.\]

Conversely, if \(gP\) is a coset, let us write \(g=hap\) with \(h\in H\), \(a\in T_P\) and \(p\in P\), and let us estimate \(d(gP, h(aPa^{-1}\cap H))\): \[d(gP, h(aPa^{-1}\cap H)) = d(aP,aPa^{-1}\cap H) \le 2B\,.\] Therefore, we have a quasi-isometry between the adorned groups \((H,\mathbb{Q})\) and \((G,\mathbb{P})\).

If \(\mathbb{P}\) is an almost malnormal collection in \(G\), let us show that \(\mathbb{Q}\) is also an almost malnormal collection in \(H\). Consider \(Q_1,Q_2\in\mathbb{Q}\), \(h\in H\) and let us assume that \(hQ_1h^{-1}\cap Q_2\) is infinite. We may find \(P_1,P_2\in \mathbb{P}\), \(a_1\in T_{P_1}\) and \(a_2\in T_{P_2}\) such that \(Q_j= a_jP_ja_j^{-1}\cap H\) for \(j=1,2\). Thus \[hQ_1h^{-1}\cap Q_2 = a_2[ (a_2^{-1}ha_1)P_1 (a_2^{-1}ha_1)^{-1}\cap P_2]a_2^{-1}\cap H\] so the almost malnormality of \(\mathbb{P}\) implies that \(P_2=P_1=P\in\mathbb{P}\) and \(a_2^{-1}ha_1\in P\). Hence, we may find \(p\in P\) such that \(h a_1= a_2p\). By definition of \(T_P\), this implies that \(a_1=a_2=a\in T_P\). Hence \(Q_1=Q_2=Q\). Furthermore, we have \(a^{-1}ha\in P\), hence \(h\in aPa^{-1}\cap H=Q\). \(\rule{.55em}{.55em}\)

3.5 Quasi-isometries between adorned groups↩︎

Let us use the results of the previous section to study quasi-isometries between adorned group.

Proof. Let \(\varphi:(G,\mathbb{P})\to (G',\mathbb{P}')\) be a quasi-isometry. Up to increasing the additive constant if necessary, we may assume that, for any \(g\in G\) and \(P\in\mathbb{P}\), there are \(g'\in G'\) and \(P'\in\mathbb{P}'\) such that \(\varphi: gP\to g'P'\) is a quasi-isometry, cf. Fact 28.

Set \(\Phi:\mathcal{C}\!\it us\,(G,\mathbb{P})\to \mathcal{C}\!\it us\,(G',\mathbb{P}')\) by letting \(\Phi=\varphi\) on \(\mathcal{C}\!\it ay\,(G)\) and extending \(\varphi\) by Fact 30 to \(\Phi:H_{gP}\to H_{g'P'}\) for any \(g\in G\) and \(P\in\mathbb{P}\). We note that there are constants \(\lambda,c\) such that each extension is a \((\lambda,c)\)-quasi-isometry on the corresponding horoball, and that \(\varphi\) is also a \((\lambda,c)\)-quasi-isometry for the graph metric of \(\mathcal{C}\!\it ay\,(G)\). Since the cosets are coarsely non-nested, it follows that \(\Phi\) is quasi-surjective.

We conclude with a subdivision argument as in the proof of [75]. Let \(x,x'\in \mathcal{C}\!\it us\,(G,\mathbb{P})\) and let us consider a geodesic \([x,x']\). We may find a subdivision \((x_j), {0}\le j\leq p\), of the geodesic segment such that each segment \([x_j,x_{j+1}]\) is non degenerate (hence have length at least \(1\)) and is either contained in \(\mathcal{C}\!\it ay\,(G)\) or in some \(H_{g_jP_{k_j}}\). Then we have \(p\leq d(x,x')\) and \[d(\Phi(x),\Phi(x'))\le \sum d(\Phi(x_j),\Phi(x_{j+1})) \le \sum (\lambda d(x_j,x_{j+1})+c) \le {(\lambda+c)d(x,x')}\,.\]

By symmetry, we obtain the same inequality for its quasi-inverse, hence Fact 27 concludes the proof (condition \((2)\) is satisfied by construction since \(\Phi_{|G}:G\to G'\) and \(\Phi_{|H_{gP}}:H_{gP}\to H_{g'P'}\) are quasi-isometries). \(\rule{.55em}{.55em}\)

If \(K\) is a geometrically finite Kleinian group, then \(K\) is finitely generated, and relatively hyperbolic with respect to its maximal parabolic subgroups [77]. Let \(\mathbb{P}\) be a set of representatives of their conjugacy classes. We may then define \(\mathcal{C}\!\it us\,(K){=\mathcal{C}\!\it us\,(K,\mathbb{P})}\) to be the cusped space of \(K\) by adding horoballs to the orbits of the elements of \(\mathbb{P}\).

Proof. By assumption, we have \(\Phi(k)=k(o)\) for any \(k\in K\). We extend \(\Phi\) to each horoball as in Fact 31. Explicitly, we proceed as follows.

Let \(U\) be a union of \(K\)-invariant horoballs in \(\mathbb{H}^3\) attached to the parabolic points of \(K\). Using Margulis’ lemma we may assume that they are pairwise disjoint (see also [78]) and that \(o\not\in U\) by decreasing their size. Fix \(P\in \mathbb{P}\) and denote by \(z_P\in\widehat{\mathbb{C}}\) its parabolic fixed point. Represent \(\mathbb{H}^3\) by the upper half-space model \(\mathbb{R}^2\times\mathbb{R}_+^*\) so that \(o\) becomes \((0,0,1)\) and \(z_P\) corresponds to the point at infinity. Thus, \(\Phi(P)\subset \mathbb{R}^2\times\{y_0\}\), where \(y_0=1\). Let \(y_1\) be large enough so that \(\mathbb{R}^2\times\{2y_1\}\subset U\) and set \(y_n=y_1\) for \(n\geq 1\). Let \(p:\mathbb{R}^2\times\mathbb{R}_+^*\to \mathbb{R}^2\) be the projection along the third coordinate and define \(\Psi_P: H_P\to \mathbb{H}^3\) by \(\Psi_P(k,n)= (p\circ\Phi(k),y_n 2^n)\). Now define \(\Phi\) on \(H_P\) by \(\Phi(k,n)=\Psi_P(k,n)\). We do the same construction for each parabolic subgroup \(P\in \mathbb{P}\) and we extend the result equivariantly to get a map \(\Phi: \mathcal{C}\!\it us\,(K)\to Hull(\Lambda_K)\). The next Claim will conclude the proof of Proposition 35. \(\rule{.55em}{.55em}\)

The map \(\Phi: \mathcal{C}\!\it us\,(K)\to \rm Hull(\Lambda_K)\) is a quasi-isometry.

Proof. Up to taking smaller horoballs that define \(U\), we may assume that for any parabolic subgroup \(P\in\mathbb{P}\) and any \(k\in P\), \(\Phi(k,1)\subset\partial U\). Let \(K_+\subset \mathcal{C}\!\it us\,(K)\) be the union of \(K\) and the vertices at height \(1\) in the horoballs and denote by \(Y_K\) the maximal induced subgraph of \(\mathcal{C}\!\it us\,(K)\) with vertex set \(K_+\). We denote by \(Z_K\) the maximal subgraph of \(\mathcal{C}\!\it us\,(K)\) whose vertices have height at least \(1\) in some horoball. Thus we have \(\mathcal{C}\!\it us\,(K)=Y_K\cup Z_K\) and \(Y_K\cap Z_K\) is the maximal subgraph of \(\mathcal{C}\!\it us\,(K)\) whose vertices have height exactly \(1\) in some horoball.

Since \(K\) is geometrically finite, the actions of \(K\) on \(\overline{\rm Hull(\Lambda_K)\setminus U}\) is cocompact [48] and, by construction, its action on \(Y_K\) is cocompact. It follows that the restriction of \(\Phi\) to \(Y_K\) is a quasi-isometry between \(Y_K\) and \(\overline{\rm Hull(\Lambda_K)\setminus U}\) endowed with the induced length metrics.

Let \(U_i\) be a horoball defining \(U\) and represent \(\mathbb{H}^3\) by the upper half-space model \(\mathbb{R}^2\times\mathbb{R}_+^*\) so that \(U_i=\{(x,y,z),z\geq 2y_1\}\). Then \(\rm Hull(\Lambda_K)\cap U_i\) has the form \(\{\rm Hull(\Lambda_K)\cap\partial U_i\}\times[2y_1,\infty)\). It follows from the construction and the previous paragraph that \(\Phi (Y_K\cap Z_K)\) lies at bounded Hausdorff distance from \(\rm Hull(\Lambda_K)\cap\partial U\). By Fact 28, \(\Phi(Y_K\cap Z_K)\) and \(\rm Hull(\Lambda_K)\cap\partial U\), each equipped with the induced path metric, are quasi-isometric and by Fact 31, for any horoball \(H_{kP}\), the restriction of \(\Phi\) to \(H_{kP}\cap Z_K\) is a quasi-isometry to the corresponding component of \(\rm Hull(\Lambda_K)\cap U\).

In the previous paragraphs, we have seen that the restrictions \(\Phi_{|Y_K}:Y_K\to \overline{\rm Hull(\Lambda_K)\setminus U}\) and \(\Phi_{|Z_K}:Z_K\to \rm Hull(\Lambda_K)\cap U\) are quasi-isometries. It follows that there is \(D\) such that the \(D\)-neighbourhood of \(\Phi(\mathcal{C}\!\it us\,(K))\) covers \(\rm Hull(\Lambda_K)\).

Pick \(k,k'\in \mathcal{C}\!\it us\,(K)\). With a subdivision as in the end of the proof of Theorem 34 we get: \[d(\Phi(k),\Phi(k'))\le 3\max\{\lambda,c\}(d(k,k')+1)\,.\]

On the other hand, we may decompose \([\Phi(k),\Phi(k')]\) into finitely many segments \([x_j,x_{j+1}]\), \({0}\leq j< p\) so that \(x_0=\Phi(k)\), \(x_p=\Phi(k')\), \(x_j\in\partial U\) for \(0< j<p\) and \((x_j,x_{j+1})\cap \partial U=\emptyset\) for \(0\leq j<p\). For each index \(0<j<p\), there is \(k_j\in Y_K\cap Z_K\) so that \(d(x_j,\Phi(k_j))\le c\) and we have \(\frac{1}{\lambda} d(k_j,k_{j+1})\leq d(x_j,x_{j+1})+3c\) for any \(j<p\), taking \(k_0=k\) and \(k_p=k'\). Let \(T>0\) be the minimal distance between two horoballs in the family \(U\). If \((x_j,x_{j+1})\cap U=\emptyset\), then \(d(x_j,x_{j+1})\geq T\) and \(d(x_j,x_{j+1})+6c\leq (\frac{6c}{T}+1)d(x_j,x_{j+1})\). Since there are at least \(\lfloor\frac{p}{2}\rfloor\) such segments, we have (shifting in the sum the constant \(3c\) of segments off \(U\) to a neighboring segment in \(U\)):

\[\begin{gather} d(k,k')\leq \sum d(k_j,k_{j+1})\leq \lambda\sum (d(x_j,x_{j+1})+3c)\\ \leq 3c\lambda+ \lambda\left(\frac{6c}{T}+1\right)\sum d(x_j,x_{j+1})=\lambda\left(\frac{6c}{T}+1\right) d(\Phi(k),\Phi(k'))+3c{\lambda}\,. \end{gather}\]

\(\rule{.55em}{.55em}\)

3.6 Quasiconvexity↩︎

Let \(X\) be a proper, geodesic, hyperbolic metric space. A \(K\)-quasiconvex subset \(Y\subset X\) has the property that any geodesic segment joining two points of \(Y\) remains in the \(K\)-neighborhood of \(Y\). Note that quasiconvexity is a property invariant under quasi-isometries.

Given a non trivial compact subset \(\Lambda\subset \partial X\), we define its weak convex hull (or join) \({\mathcal{C}}(\Lambda)\subset X\) as the union of all geodesics joining pair of points of \(\Lambda\). Even though this set is usually not convex, it is shown in [79] that it is uniformly quasiconvex. Note also that if \(\Lambda\) is a compact subset of \(\widehat{\mathbb{C}}\), then the inclusion map \({\mathcal{C}}(\Lambda)\hookrightarrow \rm Hull(\Lambda)\) is a quasi-isometry in \(\mathbb{H}^3\).

A subgroup \(H\) of a hyperbolic group \(G\) is quasiconvex if \(H\) is quasiconvex in any locally finite Cayley graph \(\mathcal{C}\!\it ay\,(G)\) of \(G\). A subgroup \(H\) of a relatively hyperbolic group \((G,\mathbb{P})\) is relatively quasiconvex if, for any cusped space \(\mathcal{C}\!\it us\,(G,\mathbb{P})\) for \(G\), there is a constant \(L\) such that for any geodesic \(\gamma\subset \mathcal{C}\!\it us\,(G,\mathbb{P})\) with endpoints in \(H\), \(\gamma\cap\mathcal{C}\!\it ay\,(G)\) remains at distance at most \(L\) from \(H\).

Moreover, according to [71], if \(H\) is a relatively quasiconvex subgroup of \(G\) and \(G\) has a cusp uniform action on \(\mathcal{C}\!\it us\,(G,\mathbb{P})\), then either \(H\) is finite or \(H\) is parabolic or the action of \(H\) admits a cusp-uniform action on some hyperbolic proper geodesic metric space which is quasi-isometric to \({\mathcal{C}}(\Lambda_H)\). In the latter case, the maximal parabolic subgroups of \(H\) define a finite number of \(H\)-conjugacy classes and are of the form \(gPg^{-1}\cap H\) where \(g\in G\), \(P\in\mathbb{P}\), and such that \(gPg^{-1}\cap H\) is infinite. It follows that \(H\) is hyperbolic relative to representatives of theses classes [71]. Moreover, its limit set \(\Lambda_H\subset\partial\mathcal{C}\!\it us\,(G,\mathbb{P})\) is also the set of accumulation points of the image \(H\hookrightarrow (\mathcal{C}\!\it us\,(G,\mathbb{P})\sqcup \partial\mathcal{C}\!\it us\,(G,\mathbb{P}))\).

We first establish some lemmas for its proof. Let us start with a geometric lemma.

Proof. Let \(z_0\in N_D({\mathcal{C}}(\Lambda))\cap N_D(Y)\). By definition, there exists \(z_1\in {\mathcal{C}}(\Lambda)\) at distance at most \(D\) from \(z_0\). If ever \(z_1\in Y\), then we are done. Otherwise, \(z_1\) belongs to a horoball \(H=H_w(p,r)\) centred at some point \(p\in\partial X\). Let us consider a geodesic \((a,b)\) with endpoints in \(\Lambda\) that contains the point \(z_1\). As \(z_1\in H\), we may find \(x \in [a,z_1)\) and \(y\in [b,z_1)\) such that \(x,y\in \partial H\cup \{p\}\). Since \(H\) is a horoball, we may assume that \(x\in X\).

Since \(z_1\in H\), then \(d(z_1,\partial H)= d(z_1,Y)\le 2D\). Let \(z_2\in\partial H\) realize this distance. It follows from [1] that \(\beta_p(.,w)\) is roughly Lipschitz, i.e., it holds \(|\beta_p(x,w)-\beta_p(y,w)|\leq d(x,y)+O(\delta)\). In particular, since \(x\) and \(z_2\) belong to \(\partial H\), we have \(|\beta_p(x,w)-r|\leq O(\delta)\) and \(|\beta_p(z_2,w)-r|\leq O(\delta)\). As \(\beta_p\) is almost a cocycle according to [1], we get \(\beta_p(x,z_2)=O(\delta)\) and \[\beta_p(x,z_1) = \beta_p(x,z_2) + {\beta_p(z_2,z_1)} +O(\delta) \le d(z_1,z_2) +O(\delta) \le 2D +O(\delta)\,.\]

If \(y\) is at infinity, then we have \(y=b=p\). In this case \(\beta_p(x,z_1)= d(x,z_1)\), and, as \(x\in {\mathcal{C}}(\Lambda)\cap Y\), we obtain \[d(z_0, {\mathcal{C}}(\Lambda)\cap Y) \le d(z_0,x)\le d(z_0,z_1) + d(z_1,x)\le 3D + O(\delta)\] Let us now assume that \(y\in X\) as well. Since triangles are slim, we may find two rays \([x,p)\) and \([y,p)\) so that the triangle \(\{x,y,p\}\) is \(\delta\)-slim. It follows that the point \(z_1\) is \(\delta\)-close to one of those rays. Let us assume for the sake of simplicity that it is \([x,p)\). It follows that \(2D\ge \beta_p(x,z_1) \ge d(x,z_1) + O(\delta)\,.\) Therefore, as \(x\in {\mathcal{C}}(\Lambda)\cap Y\), we obtain \(d(z_0, {\mathcal{C}}(\Lambda)\cap Y)\le 3D + O(\delta)\) and the lemma follows.\(\rule{.55em}{.55em}\)

We analyse relatively quasiconvex subgroups in the vicinity of their parabolic subgroups.

Proof. Let us first notice that \(a\) is also parabolic for \(G\) as \(H\) is a subgroup of \(G\). Consider a sequence \((x_n)_n\subset N_D(Ho)\cap N_D(Po)\) and \(h_n\in H\) and \(p_n\in P\) such that \(d(h_n(o),x_n)\leq D\) and \(d(p_n(o),x_n)\leq D\). We are going to show that \((h_n(o))_n\) lies in a bounded neighbourhood of \(Qo\), the conclusion follows easily.

Since \(d(p_n^{-1}h_n(o),o) \le 2D\) and the action of \(G\) is properly discontinuous, we may assume that there exists \(g\in G\) such that \(h_n=p_ng\) for all \(n\). If \((h_n)_n\) has finitely many elements, the conclusion is straightforward so we may assume the contrary.

We note that \((p_n)_n\subset P\) contains infinitely many distinct elements and the action of \(P\) on \(\partial X\setminus \{a\}\) is properly discontinuous since \(\Lambda_P=\{a\}\) as \(P\) is parabolic with fixed point \(a\), cf. [60] and [68]. Therefore, for any \(z\in \partial X\setminus\{g^{-1}(a)\}\), the sequence \((p_ng(z))_n=(h_n(z))_n\) tends to \(a\). Similarly, \((h_n^{-1}(z))_n\) tends to \(g^{-1}(a)\) for all \(z\in \partial X\setminus\{a\}\).

Since \(Q\) is a bounded parabolic subgroup of \(H\), there exists a compact subset \(K\subset \Lambda_H\setminus\{a\}\) that meets every \(Q\)-orbit in \(\Lambda_H\setminus\{a\}\). Let us pick \(z\in\Lambda_H\setminus \{g^{-1}(a)\}\). Since \(h_n(z)\in \Lambda_H\setminus\{a\}\), we may find a sequence \((q_n)\) in \(Q\) such that \(q_n(h_n(z))\in K\). If the sequence \((q_nh_n)_n\) is finite then \(d(h_n(o),Qo)\) is bounded and we are done.

Let us assume that \((q_nh_n)_n\) is infinite. Up to passing to a subsequence, we may assume that \((q_nh_n)\) is a collapsing sequence, i.e., there are two points \(\alpha,\beta\in\Lambda_H\) such that \((q_nh_n)_n\) tends uniformly to \(\beta\) on the compact subsets of \(\partial X\setminus\{\alpha\}\) [66]. Then \((q_nh_n)^{-1}_n\) tends uniformly to \(\alpha\) on the compact subsets of \(\partial X\setminus\{\beta\}\). Since \(a\) is not conical, the limit of \(((q_nh_n)^{-1}(a))_n\) has to be \(\alpha\), hence \(\alpha=g^{-1}(a)\) since \((q_nh_n)^{-1}(a)=g^{-1}(a)\) holds for all \(n\). It then follows that \(\beta\in K\) since \(z\ne \alpha\). Since \((q_np_n)_n\) is an infinite sequence in \(P\), which is a parabolic subgroup with fixed point \(a\), then \((q_nh_n(a))_n=(q_np_n(g(a)))_n\) tends to \(a\). By construction \(a\not\in K\), hence \((q_nh_n(a))_n\) can not tend to \(\beta\in K\). It follows that \(a=\alpha\). Therefore, we conclude that \(a=\alpha=g^{-1}(a)\) and \(g\in P\). It follows that \(h_n\in H\cap P=Q\) for all \(n\). Applying these arguments on subsequences, we get that for any subsequence such that \((q_nh_n)_n\) is infinite, there is a further subsequence such that \(h_n\in H\cap P=Q\). Thus we have proved that either \((q_nh_n)_n\) is finite or there are only finitely many indices such that \(h_n\not\in Q\) and the conclusion follows. \(\rule{.55em}{.55em}\)

. Let \(j=1,2\) and let us denote by \(\mathcal{C}\!\it ay\,_j\) a locally finite Cayley graph of \(G_j\) used to build the cusped space \(\mathcal{C}\!\it us\,_j=\mathcal{C}\!\it us\,(G_j,\mathbb{P}_j)\). As mentioned above, the action of \(H_j\) on \({\mathcal{C}}(\Lambda_{H_j})\;(\subset \mathcal{C}\!\it us\,_j)\) is cusp uniform. Hence the action of \(H_j\) on the truncated space \({\mathcal{C}}_T(H_j)= {\mathcal{C}}(\Lambda_{H_j})\cap \mathcal{C}\!\it ay\,_j\) is cocompact. It follows that the Hausdorff distance between \(H_j\subset \mathcal{C}\!\it us\,_j\) and \({\mathcal{C}}_T(H_j)\) is bounded since they are both \(H_j\)-invariant.

By the shadowing lemma (Lemma 2), \(\Phi({\mathcal{C}}(\Lambda_{H_1}) )\) is quasiconvex, at bounded distance from \({\mathcal{C}}(\Lambda_{H_2})\). By assumption, \(\Phi(\mathcal{C}\!\it ay\,_1)=\varphi(\mathcal{C}\!\it ay\,_1)\) is at bounded distance from \(\mathcal{C}\!\it ay\,_2\). It follows then from the previous paragraph that there is some \(D>0\) such that \(\Phi({\mathcal{C}}_T(H_1))= \varphi({\mathcal{C}}_T(H_1)) \subset N_D(\mathcal{C}\!\it ay\,_2)\cap N_D({\mathcal{C}}(\Lambda_{H_2}))\). By Lemma 3, we obtain that it is contained in a neighborhood of \({\mathcal{C}}_T(H_2)\). Thus \(\Phi({\mathcal{C}}_T(H_1))\) is at bounded distance from \({\mathcal{C}}_T(H_2)\) as well, implying that \(\Phi(H_1)\) is at bounded distance from \(H_2\) in \(\mathcal{C}\!\it us\,_2\). Let us note that the canonical injection \(\mathcal{C}\!\it ay\,_j\hookrightarrow \mathcal{C}\!\it us\,_j\), \(j=1,2\), is a coarse embedding by Fact 26, so we may conclude that \(\varphi(H_1)\) is at bounded distance from \(H_2\) in \(\mathcal{C}\!\it ay\,_2\).

It remains to prove that the quasi-isometry preserves the parings.

Let \(Q_1\in\mathbb{Q}_1\) be a parabolic subgroup of \(H_1\) with parabolic point \(p\). There are \(g_1\in G_1\) and \(P_1\in\mathbb{P}_1\) such that \(Q_1= g_1P_1g_1^{-1}\cap H_1\). By assumption, we may find \(P_2\in\mathbb{P}_2\) and \(g_2\in G_2\) such that \(\varphi(g_1P_1)\) is at bounded distance from \(g_2P_2\). Let us consider \(Q_2= g_2P_2g_2^{-1} \cap H_2\). The point \(q=\partial\Phi(p)\) is a parabolic for \(G_2\) with stabilizer \(g_2P_2g_2^{-1}\). But the point \(q\) is also a bounded parabolic for \(H_2\) since \(H_2\) is geometrically finite, so \(Q_2\) is infinite.

Let us observe that, for \(j=1,2\), \(g_jP_jg_j^{-1}\) is parabolic, so preserves horospheres centered at \(p\) or \(q\), and \(g_jP_j\) is also a horosphere centered at the same point, so \(g_jP_j\) and \(g_jP_jg_j^{-1}\) lie at bounded distance. On the one hand, \(\varphi(g_1P_1)\) is at bounded distance from \(g_2P_2\), and \(\varphi(H_1)\) is at bounded distance from \(H_2\). It follows that \(\varphi(Q_1)\) is in a bounded neighborhood of \(H_2\), of \(g_2P_2\) and \(g_2P_2g_2^{-1}\). But \(H_2\cap g_2 P_2 g_2^{-1} = Q_2\), so Lemma 4 now implies that \(\varphi(Q_1)\) is in a bounded neighbourhood of \(Q_2\). Therefore, \(\varphi\) maps the peripheral structure for \(H_1\) into a bounded neighborhood of the peripheral structure of \(H_2\). By symmetry, we conclude that \((H_1,\mathbb{Q}_1)\) and \((H_2,\mathbb{Q}_2)\) are quasi-isometric as pared groups. \(\rule{.55em}{.55em}\)

Let \((G,\mathbb{P}_G)\) be a relatively hyperbolic group and let \(\mathbb{P}\supset \mathbb{P}_G\) be a paring for \(G\) by infinite relatively quasiconvex subgroups (by definition \(\mathbb{P}\) is an almost malnormal finite collection of subgroups). Define on the Bowditch boundary \(\partial_{\mathbb{P}_G} G=\partial \mathcal{C}\!\it us\,(G,\mathbb{P}_G)\) an equivalence relation \(\sim_\mathbb{P}\) as follows: let \(x\sim_\mathbb{P}y\) if, either \(x=y\) or if there is a subgroup \(P\in \mathbb{P}\) and an element \(g\in G\) such that \(\{x,y\}\subset g(\Lambda_{P})\). Set \(Q_\mathbb{P}=\partial_{\mathbb{P}_G} G/\sim_\mathbb{P}\) to be the quotient of \(\partial_{\mathbb{P}_G} G\) by this relation \(\sim_\mathbb{P}\).

Being unable to find the following generalization of [70] in the literature, we sketch a proof of it.

We start with the following lemma.

Proof. Let \(X=\mathcal{C}\!\it us\,(G,\mathbb{P}_G)\), and let us consider the action of \(G\) on \(X\cup \partial X\). Let us consider \(H\in \mathbb{P}\setminus \mathbb{P}_G\). We note that \(H\) has no parabolic subgroups. If this was the case, then there would be some \(g\in G\) and \(P \in \mathbb{P}_G\) such that \(gPg^{-1}\cap H\) would be infinite, which contradicts the almost malnormality assumption. Therefore, since \(H\) is relatively quasiconvex, this implies that every point in \(\Lambda_H\) is conical. We may then deduce that the action of \(H\) is uniform on \(\Lambda_H\) by [69] and cocompact on \(\partial_{\mathbb{P}_G}G\setminus \Lambda_H\) by [80]. Furthermore, by the corollary of the main theorem in [80], \(H\) has finite index in \(\rm Stab_G \Lambda_H\). If \(g\in \rm Stab_G\Lambda_H\), then \(gHg^{-1} \cap H\) is a finite index subgroup of \(H\), hence infinite, so that \(g\in H\) by almost malnormality. We have proved (1) and (2).

By the corollary to [80], we also know that, for any \(K\in \mathbb{P}\setminus \mathbb{P}_G\), we have \(g\Lambda_H\cap \Lambda_K= \Lambda_{gHg^{-1}\cap K}\) so that either \(H=K\) and \(g\in H\) or \(g\Lambda_H\cap \Lambda_K=\emptyset\). This shows that the elements of \({\mathcal{K}}\) are pairwise disjoint. Let us fix \(\delta>0\). There exists \(R>0\) such that \(d(e,gH)\le R\) whenever \(\rm diamg\Lambda_H\ge \delta\) since \(H\) is quasiconvex in \(X\) [80]. This implies that there are only finitely many such elements \(g\in G/H\). This concludes the proof. \(\rule{.55em}{.55em}\)

. We proceed in two steps. We first establish that \(\sim_\mathbb{P}\) defines an upper semi-continuous decomposition of \(\partial_{\mathbb{P}_G} G\), i.e., the equivalence relation \(\sim_\mathbb{P}\) is closed. This implies that \(Q_\mathbb{P}\) is Hausdorff and compact. Then we prove that the action of \(G\) on \(Q_\mathbb{P}\) is geometrically finite with the prescribed parabolic subgroups.

By Lemma 5, the limit sets \(\{g\Lambda_P,\;P\in\mathbb{P}, g\in G\}\) form a null sequence of pairwise disjoint sets, so they define an upper semi-continuous decomposition of \(\partial_{\mathbb{P}_G} G\). This shows that the quotient \(Q_\mathbb{P}\) is a Hausdorff compact space, and the group \(G\) acts on \(Q_\mathbb{P}\).

Lemma 5 also implies (a) that the action of any \(P\in\mathbb{P}\setminus\mathbb{P}_G\) on \(\partial_{\mathbb{P}_G} G\setminus \Lambda_P\) is cocompact, so they define bounded parabolic groups on \(Q_\mathbb{P}\) and (b) that they are maximal parabolic subgroups.

Let us now check that all the other points are conical. If we consider such a point \(z\) with preimage \(x\in\partial_{\mathbb{P}_G} G\), then we may find distinct points \(\alpha,\beta\in\partial_{\mathbb{P}_G} G\) and a sequence of elements \((g_n)\) such that \((g_n(x))_n\) tends to \(\alpha\) while all the other sequences \((g_n(y))_n\) tend to \(\beta\). If \(\alpha\) and \(\beta\) lie in different fibers of the projection map \(\pi:\partial_{\mathbb{P}_G} G\to Q_\mathbb{P}\), then it follows that \(z\) is also conical. If they belong to a common fiber, they belong to some \(g\Lambda_P\), \(P\in\mathbb{P}\setminus\mathbb{P}_G\), \(g\in G\) and we may as well assume that \(g\) is trivial. As \(x\notin\Lambda_P\) and the action of \(P\) on \(\partial_{\mathbb{P}_G} G\setminus \Lambda_P\) is cocompact, we may find \((h_n)_n\) in \(P\) so that \((h_ng_n)(x)\) tends to a point \(a\in \partial_{\mathbb{P}_G} G\setminus \Lambda_P\) after extracting a subsequence if necessary. By [66], we may assume that \((h_n)_n\) has the convergence property, i.e., tends to a point \(b\in\Lambda_P\) uniformly on compact subsets of \(\partial_{\mathbb{P}_G}G\setminus\{\alpha\}\) (since \((g_n(x))_n\) tends to \(\alpha\)). Since \(\beta\ne \alpha\), it follows and \((h_ng_n(y))\) tends to \(b\) for all \(y\ne x\). Thus the limits of \((h_ng_n(x))\) and \((h_ng_n(y))_m\) will be in different fibers for all \(y\ne x\). Therefore, we may also conclude that \(z\) is conical.

In conclusion, we have defined a geometrically finite action on \(Q_\mathbb{P}\) with the prescribed maximal parabolic subgroups. As the action of \(G\) is minimal on \(\partial\mathcal{C}\!\it us\,(G,\mathbb{P}_G)\), it is also the case in its quotient \(Q_{\mathbb{P}}\). Thus \((G,\mathbb{P})\) is relatively hyperbolic and by definition, \(Q_\mathbb{P}\) is equivariantly homeomorphic to the boundary of the cusped space \(\partial\mathcal{C}\!\it us\,(G,\mathbb{P})\). \(\rule{.55em}{.55em}\) !TEX root =intro.tex

4 Canonical splittings↩︎

We describe well-known splittings of manifolds and their counterparts for finitely presented groups and show the quasi-isometric invariance of those splittings. This provides the first step in the proof of Theorems 2 and 4 as described in §[outline]. We start with the definition of a graph of groups structure and see how they appear when splitting manifolds.

Let us recall that our main results are concerned with groups up to quasi-isometry, so up to finite index. In particular, since any manifold admits a finite covering of degree at most two that is orientable, we may —and will always— assume that our manifolds are orientable.

4.1 Graph of groups and manifold splittings↩︎

Let \(G\) be a group. We first recall the definition of a graph of groups and set up some notations. We follow Serre, and define a graph as a pair of sets \((V, E)\), with a fixed-point free involution \(e\to\bar e\) on \(E\) (exchanging an oriented edge with its reverse orientation), and a terminal map \(t:E\to V\) mapping an edge to the vertex it is oriented to.

A graph of groups \({\mathcal{G}}=(\Gamma,\{G_v\},\{G_e\}, G_e\hookrightarrow G_{t(e)})\) is

1. a graph \(\Gamma=(V,E)\);

2. an assignment of a group \(G_e\) or \(G_v\) to each edge \(e\) or vertex \(v\) of \(\Gamma\), satisfying \(G_{\bar e} = G_e\); and

3. for each edge \(e \in E\), a monomorphism \(G_e\hookrightarrow G_{t(e)}\).

A graph of groups as above defines a group \(G\) up to isomorphism called the fundamental group of the graph of groups [81]. It is characterized by an action on a simplicial tree \(T\), called the Bass-Serre tree of the graph of groups, with the following properties. The action has no edge inversions and the orbit space \(T/G\) is isomorphic to \(\Gamma\). Moreover, for any vertex \(v\in T\), \(\rm Stab(v)\) is isomorphic to \(G_{p(v)}\), for any edge \(e\in T\), \(\rm Stab(e)\) is isomorphic to \(G_{p(e)}\) and the canonical injection \(\rm Stab(e)\subset \rm Stab(t(e))\) projects to the injection \(G_{p(e)} \hookrightarrow G_{t(p(e))}\), where \(p:T\to \Gamma=T/G\) denotes the canonical projection.

A graph of groups structure for a group \(G\) is a graph of groups \({\mathcal{G}}=(\Gamma,\{G_v\},\{G_e\}, G_e\hookrightarrow G_{t(e)})\) together with an isomorphism between \(G\) and the fundamental group of \({\mathcal{G}}\) as defined above.

The structure is finite if \(\Gamma\) is finite and is trivial if there is a vertex group equal to \(G\). Unless otherwise stated, graph of groups structures will be assumed to be finite and non trivial. We will also only consider graph of groups structures of a group \(G\) for which the action of \(G\) on its Bass-Serre tree \(T\) is minimal, i.e., \(T\) contains no proper \(G\)-invariant subtree. We say that \(G\) splits over a subgroup \(H\) (which can be trivial) if there is a graph of groups structure for \(G\) in which \(H\) is an edge group.

Let us first notice that a splitting of a compact three-manifold \(M\) produces a graph of groups structure for its fundamental group as follows, see [82] for details.

Let \(M\) be a compact three-manifold and let \(S\) be a finite collection of disjoint non isotopic essential surfaces. We define the tree \({\mathcal{T}}_S\) dual to \(S\) as follows: vertices are lifts of the components of \(M\setminus S\) to \(\widetilde{M}\) and there is an edge between two vertices if the closures of the corresponding components intersect. The action of \(\pi_1(M)\) on \(\widetilde{M}\) by covering transformations induces an action of \(\pi_1(M)\) on \({\mathcal{T}}_S\) by isometries. This action provides \(Q=\pi_1(M)\) with a graph of groups structure \({\mathcal{G}}=(\Delta,\{Q_v\},\{Q_e\}, Q_e\hookrightarrow Q_{t(e)})\): \(\Delta={\mathcal{T}}_S/Q\); edge groups are fundamental groups of components of \(S\) and vertex groups are fundamental groups of components of \(M\setminus S\).

The components of \(M\setminus S\) are not compact and their closures might give a different decomposition. To remedy this we set up the following definition:

4.2 Splittings over finite groups↩︎

Let \(M\) be a compact three-manifold and let \(S\subset M\) be an essential surface. As explained above, the action of \(\pi_1(M)\) on the dual tree \({\mathcal{T}}_S\) gives rise to a graph of groups structure on \(\pi_1(M)\). If \(S\) is a union of pairwise non isotopic spheres and discs, then the edge groups are trivial and if furthermore \(S\) is maximal, the vertex groups are fundamental groups of irreducible and boundary irreducible three-manifolds.

With the next proposition, we show how to reverse this construction starting from a graph of groups with vertex groups corresponding to irreducible three-manifolds.

Proof. If \(G_v\) is finite, we may as well assume that \(G'_v\) is trivial. Otherwise, since \(M_v\) is orientable and irreducible, \(G'_v\) is torsion free for any vertex \(v\) (see [83] or [84]). For an edge \(e=(v,w)\), since \(G_e\) is finite, \(G_v'\cap G_e\) and \(G_w'\cap G_e\) are trivial. Consider the graph of groups \[\overline{{\mathcal{G}}}=(\Gamma,\{\overline{G}_v,\{\overline{G}_e\}, \overline{G}_e\hookrightarrow \overline{G}_{t(e)}\})\] where \(\overline{G}_v= G_v/G_v'\) and \(\overline{G}_e= \{1\}\). Let \(\overline{G}\) be the fundamental group of \(\overline{{\mathcal{G}}}\), which is a finite graph of finite groups, hence is virtually free [82] and residually finite [85].

The canonical projections \(G_v\to\overline{G}_v\) define a projection \(q:G\to\overline{G}\) such that for any vertex \(v\), \(\ker q\cap G_v=G_v'\). Since \(\overline{G}\) is residually finite, there is a finite index subgroup \(\overline{K}\) which is disjoint from any non trivial element of any vertex group \(\overline{G}_v\). Let \(\overline{Q}=\cap_{g\in \overline{G}} g\overline{K} g^{-1}\) be the normal core of \(\overline{K}\) in \(\overline{G}\) (which has finite index in \(\overline{G}\)) and let \(p:\overline{G}\to\overline{G}/\overline{Q}\) be the canonical projection. By construction, the kernel \(Q\) of \(p\circ q\) is a finite index normal subgroup of \(G\) such that \(G_v\cap Q=G_v'\) for any vertex group \(G_v\). It follows that \(Q\) has a graph of groups structure \((\Gamma',\{G_v'\})\) with trivial edge groups.

For each edge \(e=(v,w)\) of \(\Gamma'\) we proceed as follows: if \(M_v\) and \(M_w\) have non-empty boundaries we pick a disc on \(\partial M_v\) and a disc on \(\partial M_w\) and glue \(M_v\) to \(M_w\) along those discs (we may have \(M_v=M_w\)). If \(M_v\) or \(M_w\) has no boundary, we remove a ball from \(M_v\) and a ball from \(M_w\) and glue \(M_v\) to \(M_w\) along the resulting boundary spheres. We do this operation for every edge by choosing disjoint discs and balls. It is easy to deduce from van Kampen’s theorem that the fundamental group of the resulting manifold \(M\) has a graph of groups structure \((\Gamma',\{G_v'\})\) with trivial edge groups [82]. Hence \(\pi_1(M)=Q\) and we are done.

Notice that if every manifold \(M_v\) has non-empty boundary, then we have only glued along discs so \(M\) is irreducible. If \(M_v\) has more than one boundary components then we pick all the discs corresponding to adjacent edges on the same component and if \(\chi(\partial M)<0\), we choose a component with negative Euler characteristic. This will ensure the last part of the statement as follows.

Note that, by construction, every subgroup isomorphic to \(\mathbb{Z}^2\) within a vertex group is conjugated to a subgroup of the fundamental group of a boundary torus. By Seifert–van Kampen Theorem, \(\pi_1(M)\) is obtained by starting from \(G'_v=\pi_1(M_v)\) for an arbitrary initial vertex \(v\) and inductively doing an HNN extension relative to the trivial subgroup or a free product with \(G'_w\) for another vertex \(w\). In a free product \(G_1\ast G_2\) any subgroup isomorphic to \(\mathbb{Z}^2\) lies in a conjugate of \(G_1\) or \(G_2\). Similarly, in an HNN extension relative to the trivial subgroup \(G_1\ast\mathbb{Z}\) any subgroup isomorphic to \(\mathbb{Z}^2\) lies in a conjugate of \(G_1\). It follows that every subgroup of \(\pi_1(M)\) isomorphic to \(\mathbb{Z}^2\) lies in a conjugate of a vertex group \(G'_v=\pi_1(M_v)\). Therefore, if each vertex manifolds \(M_v\) is either atoroidal with \(\chi(M_v)<0\) or homeomorphic to a ball, a solid torus or a thickened torus, we have constructed \(M\) so that it is atoroidal. \(\rule{.55em}{.55em}\)

We already mentioned the fact that a compact three-manifold \(M\) is irreducible and has incompressible boundary if and only if \(\pi_1(M)\) is one-ended. Thus, when we decompose \(M\) along a maximal union of essential spheres and discs, we get a graph of groups with trivial edge groups and one-ended or finite vertex groups.

A similar decomposition has been established by Stallings for finitely generated groups [2]. If \(G\) is non-elementary and not one-ended, it splits over a finite group, i.e., \(G\) is the fundamental group of a graph of groups with finite edge groups. Such a graph of groups is called terminal if it is finite and the vertex groups are one-ended or finite. A group which has a terminal splitting is accessible. According to Dunwoody [26], finitely presented groups are accessible.

Terminal graphs of groups are invariant under quasi-isometries [10], see below. Combining Proposition 39 and Theorem 40, it suffices to prove Theorem 2 and 4 for one-ended groups.

4.3 Torus decomposition↩︎

The second splitting of \(M\) that we will use is related to its characteristic torus decomposition. The torus decomposition together with the annulus decomposition (see §4.4) compose the JSJ splitting of \(M\). For more on the history of the next statement due to Johannson and Jaco-Shalen [86], [87], see [88].

We call \(T\) the characteristic torus decomposition of \(M\). It follows from the Torus theorem [86] that a compact Haken three-manifold that contains no essential embedded torus is either atoroidal or Seifert fibered. Thus when \(M\) is Haken, we can replace (ii) with "the compactification of every component of \(M\setminus T\) either is atoroidal or else admits a Seifert fibration". It follows from the works of Thurston and Perel’man that \(T\) is also a geometric decomposition and that non-Haken three-manifolds are geometric. Conversely geometric manifolds have empty characteristic torus decompositions except the quotients of \(Sol\) [12]. Thus the uniqueness statement in Theorem 41 implies the uniqueness (up to isotopy) of the geometric decomposition if we add a minimality assumption in the spirit of property (iii).

As explained at the beginning of §4, \(T\) induces on \(\pi_1(M)=Q\) a structure of graph of groups \({\mathcal{G}}=(\Delta,\{Q_v\},\{Q_e\}, Q_e\hookrightarrow Q_{t(e)})\) where edge groups are isomorphic to \(\mathbb{Z}^2\) and vertex groups are fundamental groups of components of \(M\setminus T\).

We would like to have a similar splitting for a group quasi-isometric to \(\pi_1(M)\). Different versions of the JSJ splitting have been given for finitely generated groups (see [89] for a survey) but they tend to be different from the one given by the torus decomposition above since \(\pi_1(M)\) may also split over cyclic subgroups. Rather than using this general theory, we will follow [27] and use the quasi-isometry to construct a splitting of \(G\). For technical purposes, we define the balanced torus decomposition \(T_B\) by replacing each component of \(T\) by two disjoint parallel copies of itself and by adding a torus parallel to each torus of \(\partial M\), thus adding a thickened torus between adjacent components of \(M\setminus T\) and next to each torus of \(\partial M\).

When \(M\) has zero Euler characteristic this result follows from the work of Kapovich and Leeb [27] and Theorem 3. In the general case the conclusion can still be deduced from the arguments of [27] and Theorem 3 as we will now explain. Let us finally recall that \((\pi_1(W(v)),P(v))\) is a (non-pared) adorned group when \(W(v)\) is a Seifert piece and a pared group when it is hyperbolic.

The starting point of Kapovich and Leeb’s proof is that \(M\) is nonpositively curved on the large scale [14]. More precisely, if \(M\) has zero Euler characteristic, there exist a nonpositively curved compact three-manifold \(N\) and a bi-Lipschitz homeomorphism between their universal covers \(\widetilde{M}\) and \(\widetilde{N}\) that preserves their torus decompositions.

For manifolds with negative Euler characteristic, we can start with a stronger statement, using the arguments of [13].

Proof. We will recall some of the arguments of the proof of [13] to show that it easily extends to manifolds with negative Euler characteristic.

Since \(\partial M\neq\emptyset\) holds, \(M\) is a Haken manifold [46]. Therefore, it follows from Theorem 41 and its following paragraph that if the characteristic torus decomposition is empty then \(M\) is either hyperbolic or admits a Seifert fibration. In the latter case, \(M\) has a finite cover \(M'\) which is a circle bundle over a compact surface \(F\) (see [46]) such that \(M'\) and \(F\) are orientable. Since \(M\) has non empty boundary, this is also the case for \(F\), so that \(M'\) is a trivial bundle and it admits an \(\mathbb{H}^2\times\mathbb{E}^1\) or \(\mathbb{E}^3\) structure which projects to \(M\). It follows that \(M\) is modeled on \(\mathbb{H}^3\), \(\mathbb{E}^3\) or \(\mathbb{H}^2\times \mathbb{E}^1\) and the conclusion is obvious.

If the characteristic decomposition \(T\), and hence the balanced decomposition \(T_B\), of \(M\) is not empty, we denote by \(M_H\) the union of the atoroidal components of \(M\setminus T_B\), by \(M_E\) the union of the Euclidean components and by \(M_G\) the closure of \(M\setminus \overline{M_H\cup M_E}\). By construction, \(M_G\) is a graph manifold with non-empty boundary and, by (the proof of) [13], \(M_G\) admits a Riemannian metric with nonpositive curvature and with flat boundary. By the hyperbolization theorem, the interior of \(M_H\) admits a complete hyperbolic metric. Now, notice that the proof of [13] consists in changing the metric of hyperbolic three-manifolds only in the rank \(2\) cusps. Following exactly the same proof, we get that any flat metric on \(T\) can be extended to a nonpositively curved metric on \(M_H\). In particular, we can extend the nonpositively curved metric on \(M_G\) to a metric on \(M\) with nonpositive curvature. \(\rule{.55em}{.55em}\)

To simplify later arguments we will list some properties of the metric on \(M\) that follow easily from the construction. The characteristic tori \(T_B\) are flat and totally geodesic. The metric on the components of \(M\setminus T_B\) of Euclidean type is actually Euclidean. The metric on the other components is locally isometric to \(\mathbb{H}^3\) or \(\mathbb{H}^2\times \mathbb{E}^1\) outside a small neighborhood of the characteristic tori. The boundary \(\partial M\) can be decomposed into a union \(\partial_{\chi=0} M\) of flat totally geodesic tori and a union \(\partial_{\chi<0} M\) of convex surfaces with negative curvature. We denote by \(\widetilde{T}_B\subset {\widetilde{M}}\) the preimage of \(T_B\) under the covering projection and call geometric component the closure of a component of \(\widetilde{M}\setminus\widetilde{T}_B\).

We now explore the action of a self-quasi-isometry of \(\widetilde{M}\) following the work of Kapovich and Leeb.

Proof.

First, we define an equivalence relation on the flats of \(\widetilde{M}\) as follows: \(F_1\sim F_2\) if and only \(F_2\) lies in the \(r\)-neighborhood of \(F_1\) for some \(r>0\). By [27], a quasi-isometry \(\varphi:\widetilde{M}\to\widetilde{M}\) induces a bijection \(\Phi\) between equivalence classes. Following [27], a flat is isolated if it is a component of \(\widetilde{T}_B\cup\widetilde{\partial_{\chi=0} M}\), the preimage of \(T_B\cup\partial_{\chi=0} M\) under the covering projection \(\widetilde{M}\to M\). As observed in [27], a flat is equivalent to an isolated flat if and only if no other flat intersects it transversally. By [27], \(\varphi\) maps an equivalence class of isolated flats to a bounded neighborhood of an equivalence class of isolated flats. By construction, a connected component \(A\) of \(\widetilde{M}\setminus \widetilde{T}_B\) is Euclidean if and only if it is an equivalence class of isolated flats. It follows that the union of the Euclidean components of \(\widetilde{M}\setminus\widetilde{T}_B\) is invariant under quasi-isometry.

On the other hand a family of flats \({\mathcal{F}}\) is equivalent to the boundary of the closure of a non-Euclidean component of \(\widetilde{M}\setminus\widetilde{T}_B\) if and only if it has the following properties:

1. \({\mathcal{F}}\) is an infinite family of pairwise non-equivalent flats, and

2. Given \(F_1,F_2\in{\mathcal{F}}\), any isolated flat separating \(F_1\) and \(F_2\) is equivalent to \(F_1\) or \(F_2\), and

3. Given \(F_1\in{\mathcal{F}}\) and \(F_2\) an isolated flat not equivalent to any element of \({\mathcal{F}}\), there is \(F_3\in{\mathcal{F}}\) such that either \(F_1\) separates \(F_2\) and \(F_3\) or \(F_3\) separates \(F_1\) and \(F_2\).

By [27], these two properties are invariant under quasi-isometries. Thus we get:

Claim.— A family of flats bounding a non-Euclidean component of \(\widetilde{M}\setminus \widetilde{T}_B\) are mapped by a quasi-isometry to a family of quasi-flats at bounded distance from a family of flats bounding a non-Euclidean component of \(\widetilde{M}\setminus \widetilde{T}_B\).

Since \(M\) is compact, the closure \(\bar A\) of any component \(A\) of \(\widetilde{M}\setminus \widetilde{T}_B\) lies in a bounded neighborhood of the union of its boundary flats.

It follows then from the above claim that any non-Euclidean geometric component is mapped by a quasi-isometry within bounded distance to another non-Euclidean geometric component.

Any non-Euclidean component \(A\) of \(\widetilde{M}\setminus \widetilde{T}_B\) is biLipschitz equivalent to a manifold \(\widetilde{V}\) where \(\widetilde{V}\) is either a product \(\widetilde{F}\times \mathbb{R}\) of the universal cover \(\widetilde{F}\) of a compact hyperbolic surface with geodesic boundary and \(\mathbb{R}\) equipped with the product metric or the universal cover of a compact hyperbolic \(3\)-manifold whose toric boundary components are flat. In particular the Hausdorff distance between two different components of \(\partial \widetilde{V}\) is infinite.

From the two previous paragraphs, we get that any non-Euclidean geometric component is mapped by a quasi-isometry within bounded distance to exactly one non-Euclidean geometric component.

From the definition of the balanced torus decomposition we get that two components \(A\) and \(A'\) of \(\widetilde{M}\setminus \widetilde{T}_B\) are adjacent if and only if one of them, say \(A'\), is Euclidean and contains a flat equivalent to a flat in \(\partial A\). From the previous discussion, we deduce then that adjacency is being preserved by quasi-isometries.

By [27], the universal cover of a hyperbolic piece can not be quasi-isometric to the universal cover of a Seifert piece. We can infer that each geometric piece is mapped to another geometric piece of the same type up to bounded Hausdorff distance. In particular, hyperbolic pieces are mapped one another. Given a geometrically finite representation \(\rho:\pi_1(W)\to\mathbb{P}SL_2(\mathbb{C})\) uniformizing a compact orientable hyperbolic manifold \(W\), \(\chi(W)=0\) if and only if \(\mathbb{H}^3/\rho(\pi_1(W))\) has finite volume or equivalently if and only if its limit set \(\Lambda_\rho\) is the whole sphere at infinity. When \(\rho(\pi_1(W))\) is minimally parabolic, by definition, \(\Lambda_\rho\) is the Bowditch boundary of \(\pi_1(W)\) relative to its (conjugacy classes of) rank \(2\) Abelian subgroups. It follows that vanishing Euler characteristic is a quasi-isometry invariant of compact hyperbolic \(3\)-manifolds and that hyperbolic pieces with negative Euler characteristic are preserved as well. \(\rule{.55em}{.55em}\)

To continue the proof of Theorem 42, we need to set up a few definitions.

Suppose that \(G\) is a group and \(\rho\) is a map from \(G\) to the set of all \((K,\epsilon)\)-quasi-isometries of a metric space \(X\). We call \(\rho\) a quasi-action of \(G\) if for some constant \(L\) and all \(g_1,g_2\in G\) the quasi-isometries \(\rho(g_1g_2)\) and \(\rho(g_1)\circ\rho(g_2)\) are \(L\)-close. The quasi-action is called quasi-transitive if for some constant \(M\) all orbits \(\rho(G).x\) are \(M\)-close to \(X\). The kernel of the action \(\rho\) is the subgroup of \(G\) which consists of elements whose action on \(X\) is at bounded distance from the identity. A quasi-action is called properly discontinuous if for each bounded subset \(C\subset X\) there are only finitely many elements \(g_j\in G\) so that \(\rho(g_j)(C)\cap C\neq\emptyset\).

We say that a collection \({\mathcal{A}}\) of subsets \(A\subset X\) is quasi-invariant under the quasi-action \(\rho\) if:

1. every bounded subset \(B\subset X\) intersects only finitely many sets in \({\mathcal{A}}\);

2. any two distinct sets in \({\mathcal{A}}\) are at infinite Hausdorff distance;

3. there is a constant \(H\) such that for all \(g\in G\) and \(A\in{\mathcal{A}}\) the set \(\rho(g)(A)\) is \(H\)-Hausdorff close to another set in \({\mathcal{A}}\).

We can define the stabilizer in \(G\) of a set \(A\) in a quasi-invariant collection \({\mathcal{A}}\). It consists of all elements \(g\in G\) such that \(\rho(g)(A)\) and \(A\) are at finite Hausdorff distance (this is relevant when the sets \(A\) are unbounded as in our case of interest). Clearly the stabilizer is a subgroup of \(G\) and it is easy to define its quasi-action on \(A\). This quasi-action is properly discontinuous and quasi-transitive if the quasi-action of \(G\) on \(X\) has such properties, see [27].

. By Proposition 43, \(M\) may be endowed with a metric of nonpositive curvature. The group \(G\) is quasi-isometric to its universal cover \(\widetilde{M}\). Let \(f_1:\widetilde{M}\to G\) and \(f_2:G\to\widetilde{M}\) be quasi-isometries such that \(d(f_2\circ f_1(x),x)\leq C\) and \(d(f_1\circ f_2(y),y)\leq C\) for some \(C\ge 0\) for any \(x\in\widetilde{M}\) and any \(y\in G\). Given \(g\in G\), it is easy to see that the map \(\rho(g):\widetilde{M}\to \widetilde{M}\) defined by \(\rho(g)(x)=f_2(g f_1(x))\) is a quasi-isometry and that \(\rho\) is a quasi-transitive properly discontinuous quasi-action of \(G\) on \(\widetilde{M}\). Let \({\mathcal{T}}\) be the tree dual to \(\widetilde{T}_B\), as defined in §4.1. By Lemma 6, \(\rho(g)\) induces an automorphism \(\sigma(g)\) on \({\mathcal{T}}\) with no edge inversion (by the definition of the balanced torus decomposition, if two components \(A\) and \(A'\) of \(M\setminus T_B\) are adjacent, then one is a thickened torus and the other one is either hyperbolic or Seifert). Since \(\rho\) is a quasi-action, we obtain \(\sigma(g_1)\sigma(g_2)=\sigma(g_1g_2)\), and we get a simplicial action of \(G\) on \({\mathcal{T}}\). The quotient \(\Gamma={\mathcal{T}}/G\) is finite and the action induces a graph of groups structure \({\mathcal{G}}=(\Gamma,\{G_v\},\{G_e\}, G_e\hookrightarrow G_{t(e)})\) for \(G\). Let \(v\) be a vertex of \({\mathcal{T}}/G\). It follows from the construction that the corresponding vertex group \(G_v\) is the stabilizer of a component \(A(v)\) of \(\widetilde{M}\setminus\widetilde{T}_B\) under the quasi-action.

Without loss of generality, we may assume that there is \(x\in \widetilde{M}\) such that \(f_1(x)\) is the neutral element in \(G\). By definition, we have \(\rho(g)(x)=f_2(gf_1(x))=f_2(g)\), in particular the image of \(f_2\) is the orbit of \(x\) under the quasi-action of \(G\) on \(\widetilde{M}\). Since the quasi-action of \(G_v\) on \(A(v)\) is quasi-transitive, [27], its image \(f_2(G_v)\) is at bounded Hausdorff distance from \(A(v)\). It follows from Fact 28 that \(G_v\) is quasi-isometric to \(A(v)\), and hence to the fundamental group \(\pi_1(W(v))\) of its projection \(W(v)\subset M\). Similarly, an edge group \(G_e\) is quasi-isometric to an isolated flat and if \(e\) is adjacent to \(v\) then this flat is equivalent to a boundary component of \(A(v)\). Thus when \(W(v)\) is a thickened torus, then \(v\) has one or two adjacent edges and for each adjacent edge \(e\), \(j_e(G_e)\) is a finite index subgroup of \(G_v\) with index \(1\) or \(2\). When \(W(v)\) is not a thickened torus, the stabilizers of adjacent edge groups \(\{G_e,\;t(e)=v\}\) are also the stabilizers of the flat components of \(\partial A(v)\) and we get, from the quasi-transitivity of the quasi-action, a quasi-isometry between \((G_v,\{G_e,\;t(e)=v\})\) and \((\pi_1(W(v)),P(v))\) where \(P(v)\) denotes the collection of the fundamental groups of the tori of \(\partial\bar W(v)\) \((W(v),\partial_{\chi =0}W(v))\). As noticed in the proof of Lemma 6 the Hausdorff distance between two flat components of \(A(v)\) is infinite, hence \(\{G_e,\;t(e)=v\}\) is a non coarsely nested collection. By construction, when two vertices \(v,w\) are adjacent then \(W(v)\) and \(W(w)\) are adjacent. This ends the proof of Theorem 42. \(\rule{.55em}{.55em}\)

We conclude this section by showing that the kernel of the quasi-action \(\rho\) defined in the latter proof is finite.

Proof. It follows from above that any element of the kernel fixes \({\mathcal{T}}\) pointwise, so the kernel is in the intersection of all vertex stabilizers.

If \(v\in{\mathcal{T}}\) is a vertex of hyperbolic type, then the restriction of the quasi-action to \(G_v\) on the corresponding component \(X_{i(v)}\) of \(\widetilde{M}\setminus T_B\) extends to a genuine action on its Gromov boundary \(\partial X_{i(v)}\) as a convergence group action [68]. Note that its Gromov boundary \(\partial X_{i(v)}\) is infinite, so, as the action of \(\ker \rho\) is trivial on \(\partial X_{i(v)}\), it must be finite since the quasi-action of \(G\) is properly discontinuous.

If there are no vertex of hyperbolic type, then, since \(M\) is non-geometric, we may find three consecutive vertices \(u,v,w\in{\mathcal{T}}\) with \(u,w\) corresponding to Seifert pieces and \(v\) to a common thickened boundary flat. It follows from [27] and its subsequent paragraphs that the kernel stabilizes the fibers up to bounded Hausdorff distance of both \(X_{i(u)}\) and \(X_{i(w)}\). By definition of the torus decomposition, these fibers intersect transversely on \(X_{i(v)}\). Therefore, the quasi-orbit of a point under the kernel is also bounded on the latter, hence the kernel is finite. \(\rule{.55em}{.55em}\)

4.4 Annulus decomposition↩︎

Lastly, we will introduce the JSJ decomposition along annuli for atoroidal three-manifolds and a generalization to relatively hyperbolic groups. This will lead to splittings of groups quasi-isometric to Kleinian groups with the following properties.

By Remark 17, we may assume that \(K\) is torsion free, geometrically finite and minimally parabolic. To prove that proposition, we will build a splitting of \(K\) in two different ways. On the one hand, we will introduce the second part of the JSJ decomposition of the Kleinian manifold \(M_K\): the characteristic annulus decomposition. This decomposition is obtained by doubling the manifold along its boundary and considering the restriction to the initial manifold of the torus decomposition of the double. Thus we get a family \(A\) of annuli that cuts the manifold into pieces with some specific topological properties. As explained at the beginning of §4.1 the action of \(K=\pi_1(M_K)\) on the dual tree \({\mathcal{T}}_A\) to \(A\) induces a graph of groups structure for \(K\). For practical purposes, we will refine \(A\) into the finer balanced annulus decomposition \(B\) with its dual tree \({\mathcal{T}}_B\) (see definition below).

On the other hand, following [90] and [28], we will build a tree \({\mathcal{T}}_{\Lambda_K}\) from the topological features of the limit set \(\Lambda_K\). We will show that \({\mathcal{T}}_{\Lambda_K}\) is isomorphic to \({\mathcal{T}}_B\) (Cor. 4) and that the action of \(K\) on \(\Lambda_K\) induces the action of \(K\) on \({\mathcal{T}}_B={\mathcal{T}}_{\Lambda_K}\). We will then see that the quasi-isometry between \(G\) and \(K\) extends to a homeomorphism between their Bowditch boundaries. This will yield an action of \(G\) on \({\mathcal{T}}_{\Lambda_K}\) and hence a splitting of \(G\). Finally the quasi-isometric invariance of those splittings will provide us with the desired properties (1) to (7).

4.4.1 Annulus decomposition and JSJ tree↩︎

We start by introducing the characteristic annulus decomposition, which, together with the torus decomposition described in Section 4.3, form the JSJ decomposition defined by Johannson-Jaco-Shalen [86], [87], see also [88]. As explained above it can be defined using the torus decomposition of the manifold obtained by doubling \(M\) along its boundary, but it will be convenient to have a more straightforward definition.

We call \(A\) the characteristic annulus decomposition of \(M\). Given a component \(W\) of \(M\setminus A\), with compactification \(\overline{W}\), we set \(P_W=(P\cap W)\cup (\partial\overline{W}\setminus (W\cap\partial M))\). When \(W\) is not a solid or thickened torus, item (ii) of Theorem 46 says that \((W,P_W)\) is a pared manifold (see definition in §2.2.1). A pared \(I\)-bundle is a pared manifold \((N,P)\) such that \(N\) is homeomorphic to a product \(F\times I\) over a compact surface \(F\) by a homeomorphism that maps \(P\) into \(\partial F\times I\).

The action of \(\pi_1(M)\) on the dual tree to \(A\) induces a graph of groups structure for \(\pi_1(M)\). As previously mentioned, we will give an alternate construction of this action when \(M\) is uniformized by a Kleinian group. To simplify the identification of the two constructions we add solid tori to \(M\setminus A\) so that a component that is pared acylindrical or a pared \(I\)-bundle is only adjacent to solid tori and thickened tori. Concretely we replace each component \(A_i\) of \(A\) that does not lie in the closure of any solid torus or thickened torus component of \(M\setminus A\) by two disjoint parallel copies of itself, thus adding a solid torus to \(M\setminus A\). We call the resulting surface \(B\) the balanced annulus decomposition of \(M\), it has the following property:

.

Notice that if we set

then the balanced annulus decomposition is uniquely defined (up to isotopy) by properties (i), (ii), (ii’) and (iii’).

To give an alternate definition of \({\mathcal{T}}_B\) using the limit set \(\Lambda_K\) of a Kleinian group \(K\) uniformizing \((M,P)\), we will now, following [28], define the subsets of \(\Lambda_K\) that will be used as vertices. Notice that since \((M,P)\) is assumed to be boundary irreducible, \(\Lambda_K\) is connected by Fact 13.

Given a continuum (i.e., a connected compact set) \(X\), a point \(x\in X\) is a cut point if \(X\setminus \{x\}\) is not connected. We define an equivalence relation \({\mathcal{R}}\) on \(\Lambda_K\). Each cut point is only equivalent to itself and if \(a,b\in\Lambda_K\) are not cut points we say that \(a{\mathcal{R}}b\) if they are not separated by any cut point, i.e. for any cut point \(c\in \Lambda_K\), \(a\) and \(b\) lie in the same component of \(\Lambda_K\setminus\{c\}\). By [28], when \(X\) is a Peano (i.e., locally connected) continuum the closure \(Y\) of each non singleton equivalence class is a Peano continuum without cut points.

Let \(Y\subset X\) be the closure of a non singleton equivalence class for \({\mathcal{R}}\). A pair \(\{a, b\} \subset Y\) is a cut pair if \(Y \setminus \{a, b\}\) is not connected. A nonempty subset \(A\subset Y\) with at least two points is called inseparable if no two points of \(A\) lie in different components of the complement of any cut pair. Every inseparable set is contained in a maximal inseparable set.

A finite subset \(S\) of a \(Y\) is called a cyclic subset if either \(S\) is a cut pair or there is an ordering \(S =\{s_j,\;j\in\mathbb{Z}/n\mathbb{Z}\}\), \(n\ge 3\), and continua \(M_j\subset Y\), \(j\in\mathbb{Z}/n\mathbb{Z}\), such that

1. \(M_i \cap M_{i+1} = \{s_i\}\), \(i\in\mathbb{Z}/n\mathbb{Z}\),

2. \(M_i \cap M_j = \emptyset\) whenever \(|i - j| > 1\),

3. \(\bigcup M_i = Y\).

An infinite subset in which all finite subsets of cardinality at least \(2\) are cyclic is also called cyclic. A maximal cyclic subset with at least 3 elements is called a necklace.

In [28], the authors use these subsets to construct an \(\mathbb{R}\)-tree associated to a Peano continuum called the combined tree. We introduce a simplified version of this construction when the continuum is the limit set \(\Lambda_K\) of a geometrically finite Kleinian group \(K\). Consider the set \(V\) of cut points, inseparable cut pairs, necklaces and maximal inseparable sets of \(\Lambda_K\) (inseparable cut pairs, necklaces and maximal inseparable sets are taken in the closures of non singleton equivalence classes of the relation \({\mathcal{R}}\) defined above). As we will see in Propositions 48 and 49, this set \(V\) is countable. We define a graph \({\mathcal{T}}_{\Lambda_K}\) with vertex set \(V\) by putting an edge between two vertices \(v_1,v_2\) if \(v_1\subset v_2\) as subsets of \(\Lambda_K\). Rather than showing that this graph is a simplicial tree, we will directly show that when \(K\) is torsion-free, it is isomorphic to the dual tree \(\mathcal{T}_B\) to the balanced annulus decomposition of the Kleinian manifold \(M_K\).

We conclude this subsection with a remark for the reader familiar with the work of [28]. We will see in Proposition 49 that the closure of a necklace intersects a maximal inseparable set only along an inseparable cut pair. Using [28], one can see that \({\mathcal{T}}_{\Lambda_K}\) is the combined tree associated to \(\Lambda_K\) defined on [28].

In the next sections we will describe the stabilizers of the sets constituting \(V\) and their relation with the characteristic annulus decomposition.

4.4.2 Cut points stabilizers↩︎

In this section we establish some properties of stabilizers of cut points. They will be used to prove that \({\mathcal{T}}_{\Lambda_K}\) is isomorphic to \({\mathcal{T}}_B\) as well as properties (1) and (2) of Theorem 45. For the latter we will work in the general situation of relatively hyperbolic groups acting on their Bowditch boundary. Let \((G,\mathbb{P})\) be a relatively hyperbolic group and assume that its Bowditch boundary \(\partial_\mathbb{P}G\) is connected. Then it also is locally connected [91]. Therefore cut points are bounded parabolic points [92], see also [91]. We study the dynamics of these parabolic subgroups intrinsically.

Proof. Let \(L\) be a compact subset whose \(H\)-translates cover \(X\). Every point in \(L\) admits a connected neighborhood in \(X\), hence contained in a unique element of \({\mathcal{C}}\). By compactness, this implies that only finitely many components of \({\mathcal{C}}\) intersect \(L\). This shows that \({\mathcal{C}}\) is a finite union of \(H\)-orbits. This also implies that any compact subset \(K\) of \(X\) meets at most finitely many components of \(X\), since the fact that the action is properly discontinuous on \(X\) implies that there are only finitely many elements of \(H\) that map points of \(K\) in \(L\).

Let us denote by \(C_1,\ldots, C_n\in{\mathcal{C}}\) the components that intersect \(L\). If \(C_i\) and \(C_j\) are in the same orbit, we fix \(h_{ij}\in H\) such that \(h_{ij}(C_j)=C_i\), with \(h_{ii}=id\). Given \(C=C_i\), write \(L_i= \cup h_{ij}(C_j\cap L)\) where the union is taken over the components \(C_j\) in the same \(H\)-orbit as \(C_i\). Note that \(L_i\) is a compact subset of \(C_i\), as a finite union of compact subsets of \(C_i\). If \(x\in C_i\), then we may find \(g\in H\) such that \(g(x)\in L\) by definition. If \(g(x)\in C_j\), then \(h_{ij}(gx)\in L_i\). Since \(h_{ij}g(C_i)=C_i\) by construction, we have proved that the action of \(\rm Stab_HC_i\) is cocompact on \(C_i\). Since \(C_i\) is non-compact (the point \(p\) is an accumulation point by definition), we may conclude that its stabilizer is infinite. \(\rule{.55em}{.55em}\)

A parabolic isometry \(g\) in a non-elementary Kleinian group \(K\) with connected limit set is an accidental parabolic if \(g\) stabilizes a component \({\mathcal{O}}\) of the ordinary set together with a geodesic in \({\mathcal{O}}\) equipped with its hyperbolic metric obtained from the Riemann uniformization theorem, since \({\mathcal{O}}\) is a proper open simply connected subset of the punctured Riemann sphere \(\widehat{\mathbb{C}}\).

Proof. We first assume that the point \(p\) is parabolic and that its stabilizer \(H\) contains an accidental parabolic element \(h\in K\). So let \(\gamma\subset \Omega_K\) be a hyperbolic geodesic stabilized by the parabolic isometry \(h\). Then \(\gamma\) joins the fixed point \(p\) of \(h\) to itself and \(\gamma\cup p\) is a Jordan curve separating \(\Lambda_K\). It follows that \(p\) is a cut point.

If \(h\) is not primitive, then there exists \(g\in K\) and an iterate \(k> 1\) such that \(g^k=h\). We first observe that \(g\in H\) since \(g^k\) fixes \(p\). Let \({\mathcal{O}}\) be the component of \(\Omega_K\) containing \(\gamma\). If \(g\) does not stabilize \({\mathcal{O}}\), then \(g(\gamma)\cap\gamma=\emptyset\) and since \(g\) acts as a translation on the uniformization \(\mathbb{C}\) of \(\widehat\mathbb{C}\setminus\{p\}\), \(g^i(\gamma)\cap\gamma=\emptyset\) for all \(i\ge 1\) contradicting the assumption \(g^k=h\). Thus we have \(g({\mathcal{O}})={\mathcal{O}}\). Since \(h\) stabilizes \(\gamma\), it acts as a hyberbolic isometry (for the hyperbolic metric) on \({\mathcal{O}}\). Basic hyperbolic geometry tells us that \(g\) must also act as a hyberbolic isometry on \({\mathcal{O}}\) with the same axis \(\gamma\). In particular, \(g\) is a primitive accidental parabolic.

We now establish the converse and suppose that \(p\) is a cut point. By Selberg’s lemma, we may assume that \(K\) is torsion free. Since \(p\) is a cut point, one can find two disjoint closed sets \(A\) and \(B\) of \(X= \Lambda_K\setminus\{p\}\) which covers \(X\). Note that, on \(\widehat{\mathbb{C}}\), \(\overline{A}\cap\overline{B}=\{p\}\). Therefore, by the separation theorem [93], we may find a Jordan curve \(c\) that separates a point \(a\in A\) from a point \(b\in B\) such that \(c\cap \Lambda_K= \{p\}\).

As mentioned at the beginning of § 4.4.2, it follows from [91] and [92] that the point \(p\) is parabolic; denote by \(H\) its stabilizer (which is isomorphic to \(\mathbb{Z}\) or \(\mathbb{Z}^2\)).

Let \({\mathcal{O}}\) be the component of \(\Omega_K\) that contains the connected set \(c\setminus\{p\}\) and \(K_{\mathcal{O}}\) be its stabilizer. Ahlfors finiteness theorem implies that \({\mathcal{O}}/K_{\mathcal{O}}\) is a surface of finite area. Moreover, Thurston proved that \(K_{\mathcal{O}}\) is geometrically finite as well, since it is finitely generated [54]. Since \(p\) is not conical for \(K\), it cannot be conical for \(K_{\mathcal{O}}\) either, so it is parabolic.

Since \(\Lambda_K\) is connected, \({\mathcal{O}}\) is simply connected so the Riemann mapping theorem provides us with a biholomorphic map \(f:\mathbb{D}^2\to{\mathcal{O}}\) from the unit disc to \({\mathcal{O}}\) and the action of \(K_{\mathcal{O}}\) on \({\mathcal{O}}\) defines a representation \(\sigma:K_{{\mathcal{O}}}\to \mathbb{P}SL_2(\mathbb{R})\). This implies that \(K_{\mathcal{O}}\cap H\) is a cyclic group. By [94], \(\partial {\mathcal{O}}\) is locally connected and by the Carathéodory-Torhorst theorem [95], \(f\) extends continuously to \(\partial\mathbb{D}^2\) (see also [96]). The curve \(c\) disconnects \(\partial{\mathcal{O}}\) since, otherwise, \(c\) would bound a disc in \({\mathcal{O}}\), and, hence, would not separate \(A\) and \(B\). It follows that \(\partial\mathbb{D}^2\setminus\{f^{-1}(p)\}\) is not connected which is only possible if \(\sharp\{f^{-1}(p)\}>1\). Then we get from [96] that \(\sharp\{f^{-1}(p)\}= 2\). Therefore the hyperbolic geodesic \(\ell\) joining the two points of \(f^{-1}(p)\) is also invariant and \(f(\ell)\subset{\mathcal{O}}\) is invariant under the action of \(K_{\mathcal{O}}\cap H\). This implies that \(p\) is an accidental parabolic. \(\rule{.55em}{.55em}\)

Proof. Let \(H\) be the stabilizer of \(p\). As \(p\) is at infinity, a parabolic isometry fixing \(p\) acts on \(\mathbb{C}\) by translations. Since \(H\) is discrete and isomorphic to \(\mathbb{Z}^2\), its action on \(\mathbb{C}\) is cocompact.

It follows from Proposition 48 that there is a component \({\mathcal{O}}\) of \(\Omega_K\) that is fixed by a cyclic group generated by a primitive element \(h_1\). As \(h_1\) is primitive, Bezout’s theorem implies the existence of a primitive element \(h_2\) such that \(H\) is generated by \(h_1\) and \(h_2\).

Moreover, \(h_2\) acts freely on the \(H\)-orbit of \({\mathcal{O}}\), and so on the components of \(\Lambda_K\cap \mathbb{C}\) since the \(H\)-orbit of \({\mathcal{O}}\) splits the plane in parallel (topological) strips in the following sense. By Proposition 48, There is an \(h_1\)-invariant curve in \({\mathcal{O}}\) that is translated by all the iterates of \(h_2\). These curves cut the plane into strips.

Let \(C\) be a component of \(\Lambda_K\cap \mathbb{C}\), and assume that \(h\in H\) fixes \(C\). Then we may find \(k,\ell\in\mathbb{Z}\) such that \(h=h_1^{k}h_2^{\ell}\). It follows from above that \(\ell=0\) and \(h\) is an iterate of \(h_1\). This shows that \(\rm Stab_HC\subset H_{\mathcal{O}}\). We claim that \(h_1(C)=C\). To see this, consider the canonical projection \(\pi:\mathbb{C}\to \mathbb{C}/(h_1^k)\) to the cylinder. The set \(C\) projects into a connected compact subset that separates both ends of the cylinder. The map \(h_1\) acts as an automorphism of order \(k\) and maps \(\pi(C)\) to \(\pi(h_1(C))\) that has the same property: if \(h_1(C)\ne C\), then \(\pi(h_1(C)))\cap \pi(C)=\emptyset\), but then \(h_1\) would not be of finite order. It follows that \(h_1(C)=C\) as claimed. Thus, every component \(C\) has stabilizer the subgroup generated by \(h_1\), i.e., \(H_{\mathcal{O}}\).

Claim.— The sequence \((h_2^n)_n\) is uniformly convergent to \(p\) on any component \(C\) of \(X\).

If not, there would be a sequence of points \((x_k)\) in \(C\) such that \(h_2^{n_k}(x_k)\) remains in a compact subset of \(X\) for some subsequence \((n_k)\). Since the action of \(\rm Stab_HC\) is cocompact on \(C\) by Proposition 47, we may assume that \((x_k)\) remains in a compact subset of \(C\) precomposing \(h_2^{n_k}\) by a suitable iterate of \(h_1\). Since the action is properly discontinuous, this implies that an iterate of \(h_2\) fixes \(C\), which is absurd. This proves the claim.

Fix \(R>0\) and a component \(C\) of \(\Lambda_K\cap\mathbb{C}\). Fix another component \(C'\), and let assume that there are \(m\) translates of \(C'\) at distance at most \(R\) from \(C\). We may find \((n_k)\), \(1\le k\le m\), such that \(\rm dist(C,h_2^{n_k}C')\le R\) for all \(k\). Since the actions of the stabilizers of components are cocompact on their components (cf. Proposition 47), we may assume that these distances are realized in some compact subset of \(X\). Together with the fact that there are finitely many \(H\)-orbits of \(X\) by Proposition 47, the claim implies that \(m\) has a uniform upper bound. So there are only finitely many components at distance at most \(R\) from \(C\).\(\rule{.55em}{.55em}\)

Proof. We may assume that the point \(p\) is at infinity in \(\widehat{\mathbb{C}}\) so that \(G\) acts by proper uniformly quasi-Möbius maps on \(\Lambda_K\setminus\{p\}\subset \mathbb{C}\) endowed with the Euclidean metric. Notice that as \(x_4\) tends to infinity, the crossratio \([x_1:x_2:x_3:x_4]\) will degenerate into \(d(x_1,x_2)/d(x_1,x_3)\), showing that \(G\) acts by uniform quasisymmetric maps on \(X= \mathbb{C}\cap\Lambda_K\). This means that there exists an increasing homeomorphism \(\eta:\mathbb{R}_+\to\mathbb{R}_+\) such that, for any \(x,y,z\in X\), any \(t>0\) and any \(g\in G\), \[if|x-y|\le t|x-z|then|g(x)-g(y)|\le \eta(t) |g(x)-g(z)|\,.\]

Since \(p\) is a cut point, \(X\) has at least \(2\) connected components. Let \(H\subset K\) denote the stabilizer of \(p\). By assumption, it is a rank \(2\) Abelian group acting properly discontinuously by translations on \(\mathbb{C}\). By Corollary 3 there are constants \(\delta_\pm >0\) such that, for any component \(C\) of \(X\), \(\rm dist(C,X\setminus C)\ge \delta_-\) and for any \(x\in C\), \(\rm dist(x,X\setminus C)\le \delta_+\) hold.

We first prove that we may deform the Euclidean metric on \(X\) quasisymmetrically so that \(G\) acts by isometries. Let \(x,y\in X\). We claim that \(d_{X}(x,y)= \sup\{|g(x)-g(y)|,\;g\in G\}\) is finite. Let \(C_x\) be the component of \(X\) containing \(x\). Given \(g\in G\), pick \(z\in X\setminus C_x\) so that \(|g(x)- g(z)|= \rm dist(g(x),X\setminus g(C_x))\), then we have \[|g(x)-g(y)| \le \eta\left(\frac{|x-y|}{|x-z|}\right) |g(x)-g(z)|\le \eta\left(\frac{|x-y|}{\delta_-}\right) \delta_+\,.\]

This implies that \((X,d_{X})\) is a metric space on which \(G\) acts by isometries. Moreover, for any \(x,y,z\in X\) and \(\varepsilon>0\), if \(g\in G\) satisfies \(d_{X}(x,y)\le (1+\varepsilon)|g(x)-g(y)|\), then \[\frac{d_{X}(x,y)}{d_{X}(x,z)}\le (1+\varepsilon) \frac{|g(x)-g(y)|}{|g(x)-g(z)|} \le (1+\varepsilon) \eta\left(\frac{|x-y|}{|x-z|}\right)\,.\] Since \(\varepsilon\) is arbitrary, it follows that \(id:(X,|\cdot|)\to (X,d_{X})\) is \(\eta\)-quasisymmetric.

Since the action of \(G\) is cocompact on \(X\) and parabolic, Proposition 47 implies that there are only finitely many orbits of such connected components. Moreover, their stabilizer are virtually cyclic since \(G\) has rank two and there are infinitely many components, cf. Corollary 3 applied to the Kleinian group \(K\).

If \(C'\) is another component, then, for any \(g\in\rm Stab_G C\), \(d_X(C,g(C'))={d_X}(C,C')\). By the definition of \(d_X\), the Euclidean distance \(d_e(C,g(C'))\) between \(C\) and \(g(C')\) is bounded from above by \(d_X(C,C')\) : \(d_e(C,g(C'))\le d_X(C,g(C'))\le d_X(C,C')\). It follows then from Corollary 3 that the orbit of \(C'\) under \(\rm Stab_G C\) is finite. This implies that \(\rm Stab_G C\cap \rm Stab_G C'\) has finite index in \(\rm Stab_G C\). Therefore, stabilizers are commensurable.

Let \(G'\) be a rank two free Abelian subgroup of finite index in \(G\). Then stabilizers of components in the same \(G'\)-orbit are the same and cyclic since they are infinite. They also act cocompactly on each of them. Therefore, there are only finitely many such stabilizers and their intersection is a cyclic group generated by some \(g_1\) of finite index in any stabilizer. Let us consider a primitive \(g_2\in G'\) so that it generates with \(g_1\) a subgroup \(G_A\) of rank two. The element \(g_2\) acts freely on the components of \(X\) by construction. \(\rule{.55em}{.55em}\)

4.4.3 Stabilizers and characteristic annulus decomposition↩︎

Next we will study the relations between the stabilizers of the sets that make up \(V\) and the balanced annulus decomposition. We start with the cut points.

Notice that, by Fact 13, \(\partial M\setminus P\) is incompressible and we can use Theorem 46 to define the characteristic annulus decomposition and then the balanced annulus decomposition \(B\).

Proof. Let \(c\in\Lambda_K\) be a cut point. By Proposition 48, \(c\) is stabilized by a parabolic subgroup \(H<K\).

Let \(\widetilde{\gamma}\subset{\mathcal{O}}_K\) be the union of all the geodesics in the ordinary set \({\mathcal{O}}_K\) that are stabilized by an element of \(H\). It follows from [96], that each component of \({\mathcal{O}}_K\) can contain at most one of those geodesics. Then \(\widetilde{\gamma}\) projects to a multicurve \(\gamma\subset\partial M\). Since \(K\) uniformizes \((M,P)\) there is a component \(P_i\) of \(P\) such that \(H\) is conjugate to \(\rho(\pi_1(P_i))\) hence each component of \(\gamma\) is homotopic to a closed curve on \(P_i\). By the Annulus theorem [86] there is an essential annulus \(E_{\gamma_k}\) joining each component \(\gamma_k\) of \(\gamma\) to \(P_i\). By Proposition 48 and Corollary 3, there is a single primitive element \(h \in H\) such that every component of \(\widetilde{\gamma}\) has \(H\)-stabilizer equal to \(\langle h\rangle\), hence all the components of \(\gamma\) are homotopic to the same simple closed curve on \(P_i\). A simple surgery argument allows us to construct a family of disjoint annuli \(E_{\gamma_k}\) joining the components of \(\gamma\) to \(P_i\). Let us denote by \(\Delta_i\subset M\) the \(2\)-complex made up of these annuli and \(P_i\) and by \(V_{P_i}\subset M\) a regular closed neighborhood of \(\Delta_i\).

The frontier \(Fr(V_{P_i})\) of \(V_{P_i}\) is a union of essential annuli. It is a fundamental property of the JSJ decomposition that any essential annulus is isotopic to an annulus that is disjoint from \(B\), see [88]. It follows then from the minimality of the characteristic annulus decomposition that \(V_{P_i}\) is contained in a component \(W_{P_i}\) of \(M\setminus B\) up to isotopy. Let us assume that \(V_{P_i}\) is not isotopic to its compactification \(\overline{W}_{P_i}\). Then there is a component \(F\) of \(Fr(V_{P_i})\) that is not peripheral in \(\overline{W}_{P_i}\), in particular \((\overline{W}_{P_i},(B\cap\overline{W}_{P_i})\cup P_i)\) is not pared acylindrical. If \(W_{P_i}\) is an \(I\)-bundle \(S\times I\) then \(P_i\) and \(F\) are union of fibers, up to isotopy, by [11]. Since \(F\) is homotopic to a curve in \(P_i\) this would contradict the fact that \(F\) is boundary incompressible and not peripheral. Thus we have proved that \(W_{P_i}\) is a solid or thickened torus, and by construction, \(c\) is the fixed point of a conjugate of \(\rho(\pi_1(W_{P_i}))\). We can further deduce from the maximality of \(\widetilde{\gamma}\) that \(\overline{W}_{P_i}\) and \(V_{P_i}\) are isotopic.

Conversely, let \(W\) be a solid torus or thickened torus in \(M\setminus B\) containing a component \(P_i\) of \(P\). Then \(\rho(\pi_1(W))\) is a parabolic subgroup stabilizing a point \(c\in\Lambda_K\). Let \(F\) be a component of \(W\cap\partial M\) which is disjoint from \(P\) and let \(\gamma\) be a simple closed curve on \(F\). The curve \(\gamma\) lifts in \(\Omega_K\) to an arc \(\widehat\gamma\) joining \(c\) to itself. By construction, \(\widehat\gamma\cup c\) separates \(\Lambda_K\). It follows that \(c\) is a cut point. \(\rule{.55em}{.55em}\)

Let \(W_P\subset M\setminus B\) be the union of the connected components \(W_i\) of \(M\setminus B\) such that \(\rho(\pi_1(W_i))\) is a parabolic subgroup and \(W_i\) is either a thickened torus or a solid torus (the latter case happens only if \(K\) is not minimally parabolic). It follows from the minimality of the balanced annulus decomposition (Property (iii’)) that no two components of \(W_P\) are adjacent. Recall that if \(a,b\in\Lambda_K\) are not cut points then \(a{\mathcal{R}}b\) if they are not separated by any cut point.

Proof.

Given a component \(W_{P_i}\) of \(W_P\), in the proof of Lemma 8, we constructed a \(2\)-complex \(\Delta_i\) so that \(W_{P_i}\) is isotopic to a neighborhood of \(\Delta_i\). Using the same notations as in the proof of Lemma 8, we will introduce another way to build \(\Delta_i\setminus P_i\) using the Kleinian group \(K\).

Each component \(\widetilde{\gamma}_k\) of \(\widetilde{\gamma}\) is stabilized by a subgroup \(H_{\widetilde{\gamma}}<H<K\) isomorphic to \(\mathbb{Z}\) and the closure \(\widetilde{\gamma}_k\cup c\) of \(\widetilde{\gamma}_k\) bounds in \(\mathbb{H}^3\cup\widehat\mathbb{C}\) a closed disc \(\widetilde{D}_{\widetilde{\gamma}_k}\) whose interior lies in \(\mathbb{H}^3\) and that corresponds to the closure of the universal covering of \(E_{\gamma_k}\) in \(\mathbb{H}^3\cup\Omega_K\approx \widetilde{M\setminus P}\). As these annuli are pairwise disjoint, these discs are also

pairwise disjoint and form a \(K\)-invariant collection. We denote by \(\widetilde{D}_{P_i}=\bigcup \widetilde{D}_{\widetilde{\gamma}_k}\) the union of all these discs. The projection \(D_{P_i}\) of \(\widetilde{D}_{P_i}\cap (\mathbb{H}^3\cup \Omega_K)\) to \((\mathbb{H}^3\cup \Omega_K)/K\approx M\setminus P\) is a family of disjoint half infinite annuli joining the components of \(\gamma\) to the cusp associated to \(P_i\). By construction \(D_{P_i}\) is equal to \(\Delta_i\setminus P_i\) as promised.

Doing the same construction for each component of \(P\) we get that every component \(V\) of \(( (\mathbb{H}^3\cup \Omega_K)/K)\setminus\bigcup_i D_{P_i}\) is isotopic to a component of \(M\setminus\bigcup_i \Delta_i\) and hence to a component \(W\) of \(M\setminus W_P\) according to the proof of Lemma 8. It follows that if \(\widehat V\subset\mathbb{H}^3\cup\widehat{\mathbb{C}}\) is the closure of a lift \(\widetilde{V}\) of a component \(V\) of \(( (\mathbb{H}^3\cup \Omega_K)/K)\setminus\bigcup_i D_{P_i}\), then \(\Lambda_K\cap\widehat V\) is the limit set \(\Lambda_{K_W}\) of a conjugate \(K_W\) of \(\rho(\pi_1(W))\).

Since \(\partial W\setminus P\) is incompressible, \(\Lambda_{K_W}\) is connected. Since \(B\cap W\) is the balanced decomposition of \((W,P\cap W)\), \(\Lambda_{K_W}\) does not have any cut point by Lemma 8. Hence \(\Lambda_{K_W}\) lies in the closure of a non-singleton equivalence class of \({\mathcal{R}}\). Given a point \(x\in\Lambda_K\setminus \Lambda_{K_W}\), any point \(y\) in \(\Lambda_{K_W}\) which is not a cut point of \(\Lambda_K\) is separated from \(x\) in \(\mathbb{H}^3\cup \widehat{\mathbb{C}}\) by a component of \(\widetilde{D}\). Hence \(x\) and \(y\) are separated in \(\widehat{\mathbb{C}}\) by the closure of a component of \(\widetilde{\gamma}\) and in \(\Lambda_K\) by a cut point. It follows that \(\Lambda_{K_W}\) is the closure of a nonsingleton equivalence class of \({\mathcal{R}}\).

By Lemma 8, any point in \(\Lambda_K\) that is not a cut point lies in \(\widehat{\mathbb{C}}\setminus \bigcup_{\widetilde{c}\in\widetilde{\gamma}} \widetilde{c}\). Thus we have obtained above all the nonsingleton equivalence classes of \({\mathcal{R}}\). \(\rule{.55em}{.55em}\)

Let \((W,P_W)\) be the compactification of a component of \(M\setminus W_P\) equipped with its induced paring. It is easy to see that the balanced annulus decomposition of \(W\) relative to \(P_W\) is \((B\cap W)\setminus P_W\). Thus, after Lemmas 8 and 9 it only remains to study the relation between the sets in \(V\) and \(B\) for the subgroup \(K_W\) of \(K\) uniformizing such a submanifold \((W,P_W)\), i.e., with connected limit set without cut points.

Consider a Kleinian group \(K\) uniformizing a pared manifold \((M,P)\) together with a homeomorphism identifying \(M\setminus P\) with \((\mathbb{H}^3\cup\Omega_K)/K\) and a representation \(\rho:\pi_1(M)\to \mathbb{P}SL_2(\mathbb{C})\). Assume that \(\Lambda_K\) is connected and let \(E\) be a connected component of \(B\) such that \(\rho(\pi_1(E))\) is generated by a parabolic isometry. Since \(K\) uniformizes \((M,P)\), \(E\) can be homotoped (without keeping its boundary on \(\partial M\)) in a component \(P_i\) of \(P\). Since \(E\) is essential and \((M,P)\) is a pared manifold, at least one component of \(\partial E\) is not homotopic to \(P_i\) on \(\partial M\). It follows that \(\rho(\pi_1(E))\) is an accidental parabolic and, by Proposition 48, that \(\Lambda_K\) has a cut point.

Thus we have proved that when \(\Lambda_K\) is connected with no cut point, for any component \(E\) of \(B\), \(\rho(\pi_1(E))\) is generated by a loxodromic isometry. In particular, since any rank \(2\) Abelian subgroup of \(K\) is parabolic, no component of \(M\setminus B\) is a thickened torus.

Let \(\mathbb{P}\) be the paring in \(K\) defined by the conjugacy classes of the parabolic subgroups. From Proposition 35 we get a quasi-isometry \(\Phi:\mathcal{C}\!\it us\,(K,\mathbb{P})\to \mathrm{Hull}(\Lambda_K)\). If \(E\) is a connected component of \(B\), since \(\rho(\pi_1(E))\) is generated by a loxodromic isometry, then \(\widetilde{E}\cap\mathrm{Hull}(\Lambda_K)\subset\mathbb{H}^3\) is quasiconvex for any connected component \(\widetilde{E}\) of the preimage of \(E\). It follows that \(\widetilde{W}\cap\mathrm{Hull}(\Lambda_K)\) is quasiconvex where \(\widetilde{W}\subset\mathbb{H}^3\cup\Omega_K\) is a component of the preimage of a component \(W\) of \(M\setminus B\). Then \(\Phi(\mathcal{C}\!\it us\,(K_{\widetilde{W}},\mathbb{P}_{\widetilde{W}}))\) lies at bounded Hausdorff distance from \(\widetilde{W}\cap\mathrm{Hull}(\Lambda_K)\) where \(K_{\widetilde{W}}<K\) is the stabilizer of \(\widetilde{W}\) and \(\mathbb{P}_{\widetilde{W}}\) is the induced paring (obtained from Definition 32 and Fact 33). From Fact 28, we deduce that \(\widetilde{W}\cap\mathrm{Hull}(\Lambda_K)\) is quasi-isometric to \(\mathcal{C}\!\it us\,(K_{\widetilde{W}},\mathbb{P}_{\widetilde{W}})\) and that the ideal boundary of \(\widetilde{W}\cap\mathrm{Hull}(\Lambda_K)\subset (\mathbb{H}^3\cup\Omega_K)\) is the limit set of \(K_{\widetilde{W}}\) which is a conjugate of \(\rho(\pi_1(W))\).

To prove Proposition 49 we use the work of Walsh [29] where the author explains the relation between the bumping sets of the connected components of \(\Omega_{K}\) and the characteristic submanifold of \(\mathbb{H}^3/K\).

First we use the separation theorem to establish the relation between the cut pairs in \(\Lambda_{K}\) and the bumping set of \(\Omega_{K}\).

Proof. Since \(\{x,y\}\) is a cut pair, one can find two closed subsets \(A\) and \(B\) of \(X\) such that \(X=A\cup B\) and \(A\cap B=\{x,y\}\). By the separation theorem [93], there is a Jordan curve \(c\) that separates a point \(a\in A\) from a point \(b\in B\) such that \(c\cap X={\{x,y\}}\). The set \(c\setminus{\{x,y\}}\) has two connected components \(k\) and \(k'\). Let us show that \(k\) and \(k'\) do not lie in the same connected component of \(\Omega\). Otherwise, we would find an arc \(k''\) in \(\Omega\) joining \(k\) and \(k'\) so that \(c\cup k''\) is a \(\theta\)-curve. Among the boundaries of the three components of its complement, the two that contain \(k''\) only intersect \(X\) at a single point. But, as \(A\cup B\) intersects at least two of the three components of the complement of \(c\cup k''\), this would imply that \(x\) or \(y\) is a cutpoint — a contradiction. The two components \({\mathcal{O}}\) and \({\mathcal{O}}'\) of \(\Omega\) containing \(k\) and \(k'\) respectively satisfy the conclusion.

Conversely, if there are two components \({\mathcal{O}}\) and \({\mathcal{O}}'\) of \(\Omega=\widehat\mathbb{C}\setminus X\) such that \(\{x,y\}\subset \overline{{\mathcal{O}}}\cap\overline{{\mathcal{O}}'}\), consider \(c=k\cup k'\cup\{x,y\}\) where \(k\subset{\mathcal{O}}\) and \(k'\subset{\mathcal{O}}'\) are the geodesics joining \(x\) to \(y\) (\({\mathcal{O}}\) and \({\mathcal{O}}'\) are Jordan domains equipped with their Poincaré metrics, i.e., the hyperbolic metrics obtained from the Riemann uniformization theorem). Since the two components \({\mathcal{C}}_1\) and \({\mathcal{C}}_2\) of \(\widehat{\mathbb{C}}\setminus c\) intersect both \({\mathcal{O}}\) and \({\mathcal{O}}'\), both \({\mathcal{C}}_1\) and \({\mathcal{C}}_2\) intersect \(X\). In particular \(c\) separates \(X\) and \(\{x,y\}\) is a cut pair. \(\rule{.55em}{.55em}\)

Using this lemma and results of [29], see also [30], we can study the stabilizer of a necklace.

Proof. Let \(X\) be a necklace and let us consider a finite cyclic subset \(S =\{s_j,\;j\in\mathbb{Z}/n\mathbb{Z}\}\), \(n\ge 3\), together with defining continua \(M_j\subset \Lambda_K\), \(j\in\mathbb{Z}/n\mathbb{Z}\). By definition, any pair of points in a necklace is a cut pair. We can argue in a similar manner to the proof of Lemma 10 to show that there are open Jordan domains \(\{D_j,\;j\in\mathbb{Z}/n\mathbb{Z}\}\) such that \(M_j\subset D_j\cup\{s_{j-1},s_j\}\), \(\partial\overline{D_j}\setminus \{s_{j-1},s_j\}\) consists of two arcs in different components of \(\Omega_K\), \(\overline{D_j}\cap \overline{D_k}\ne \emptyset\) only if \(|j-k|\le 1\) and \(\overline{D_j}\cap \overline{D_{j+1}}= \{s_j\}\). By construction, \(\widehat{\mathbb{C}}\setminus \cup_{j\in\mathbb{Z}/n\mathbb{Z}} \overline{D_j}\) has exactly two components, and since \(\Lambda_K\subset\bigcup_j M_j\), they are contained in \(\Omega_K\) . As \(D_1\setminus\Lambda_K\) lies in two different components of \(\Omega_K\), we may conclude that there are two components of \(\Omega_K\) that contain \(S\) on their boundaries. All the other components of \(\Omega_K\) are contained in one of the domains \(D_j\), so cannot contain \(S\) on their boundaries. It follows then that \(X\) lies in the frontiers of two components of \(\Omega_K\). By [29], \(X\) is the limit set of a conjugate of \(\rho(\pi_1(W))\) for an \(I\)-bundle component \(W\) of \(M\setminus B\). Conversely, for such a component \(W\) of \(M\setminus B\) which is a pared \(I\)-bundle, every pair of points in the limit set of \(\rho(\pi_1(W))\) is a cut pair. It follows then from [28] that this limit set is a necklace. \(\rule{.55em}{.55em}\)

Before explaining the proof, let us notice that since no component of \(M\setminus B\) is a thickened torus, any component of \(B\) lies in the boundary of the closure of a component of \(M\setminus B\) that is a solid torus.

Proof. By Lemma 10, \(\{a,b\}\) is a cut pair if and only if there are two components \({\mathcal{O}}\) and \({\mathcal{O}}'\) of \(\Omega_K\) such that \(\{a,b\}\subset Y= \overline{{\mathcal{O}}}\cap\overline{{\mathcal{O}}'}\). By [29] (see also [30]), the set \(Y\) is the limit set \(\Lambda_{K_W}\) of a subgroup \(K_W<K\) which uniformizes the compactification \(W\) of a component of \(M\setminus B\). By Lemma 10, every pair of points in \(Y\) is a cut pair and by [28] either \(Y\) is an inseparable cut pair or a necklace.

If \(Y\) is an inseparable cut pair, then \(\pi_1(W)\) is conjugated to the stabilizer of \(Y=\{a,b\}\). This can only happen if \(W\) is a solid torus.

If \(Y\) is a necklace, \(W\) is an \(I\)-bundle over a compact surface \(F\) by Lemma 11. Set \(P^0_W=P\cap W\) and denote by \(P_F\) the projection of \(P_W^0\) on \(F\) along the fibers. Let \(H_F\) be a Fuchsian group uniformizing \((F,P_F)\), then \(H_F<\mathbb{P}SL_2(\mathbb{R})<\mathbb{P}SL_2(\mathbb{C})\) uniformizes \((W,P_W^0)\) as a Kleinian group. On the other hand, as previously mentioned, the subgroup \(K_W\) of the Kleinian group \(K\) also uniformizes \((W,P_W^0)\). Let \((G_W,\mathbb{P}_W)\) be the pared fundamental group of \((W,P^0_W)\). Proposition 35 provides us with well-defined homeomorphisms between \(\Lambda_{H_F}\) and \(\partial\mathcal{C}\!\it us\,(G_W,\mathbb{P}_W)\) on the one hand and between \(\Lambda_{K_W}\) and \(\partial\mathcal{C}\!\it us\,(G_W,\mathbb{P}_W)\) on the other hand from which we get a homeomorphism between \(\Lambda_{H_F}\) and \(\Lambda_{K_W}\). The action of \(H_F\) on \(\mathbb{H}^2\) defines a hyperbolic metric on \(F\setminus P_F\). Since \(a\neq b\), they are joined by a geodesic in \(\mathbb{H}^2\). Let \(l\) be the projection on \(F\setminus P_F\) of this geodesic. Notice that any geodesic in \(F\) crossing \(l\) defines a cut pair in \(\Lambda_{K_W}\) separating \(a\) and \(b\) in \(\Lambda_K\). On the other hand if \(\{x,y\}\) is a cut pair separating \(a\) and \(b\), by [28], \(Y\cup\{x,y\}\) is cyclic and by maximality \(\{x,y\}\subset Y\). It follows that \(l\) is homotopic to a component of \(\partial F\setminus P_F\) if and only if \(\{a,b\}\) is inseparable. Since \(W\) is the compactification of a component of \(M\setminus B\), the closures of the components of \(\partial W\setminus\partial M\) are components of \(B\). In particular if \(l\) is homotopic to a component of \(\partial F\) then \(\{a,b\}\) is the limit set of a conjugate of \(\rho(\pi_1(E))\) for a component \(E\) of \(B\). As mentioned before the proof, any component of \(B\) is adjacent to a solid torus and the conclusion follows.

Conversely, if \(W\) is a solid torus and \(\{a,b\}\) is the limit set of \(\rho(\pi_1(W))\), consider two simple closed curves \(\gamma_1,\gamma_2\) on different components of \(W\cap\partial M\) so that \(\gamma_1\cup\gamma_2\) is isotopic on \(\partial M\) to the boundary of a component of \(B\). Then \(\gamma_1\cup\gamma_2\) bounds an essential annulus \(E\) (isotopic to the aforementioned component of \(B\)). If \(\gamma_1\cup \gamma_2\) also bounded an annulus \(E'\subset \partial M\), then, since \(M\) is atoroidal, either \(E\cup E'\) would bound a solid torus or \(E\cup E'\) would be homotopic to a surface immersed in \(\partial M\). In both cases this would contradict the fact that \(E\) is essential (in particular boundary incompressible). Therefore \(\gamma_1\cup \gamma_2\) does not bound an annulus in \(\partial M\).

If \(\widetilde{\gamma}_1\) and \(\widetilde{\gamma}_2\) are lifts of \(\gamma_1, \gamma_2\) with endpoints \(a\) and \(b\), then the closure of \(\widetilde{\gamma}_1\cup\widetilde{\gamma}_2\) separates \(\widehat{\mathbb{C}}\). If they were in the same component of \(\Omega_K\), the curves \(\gamma_1\) and \(\gamma_2\) would be homotopic within \(\partial M\), contradicting the fact that they do not bound an annulus in \(\partial M\), so \(\widetilde{\gamma}_1\) and \(\widetilde{\gamma}_2\) lie in different components of \(\Omega_K\). Hence \(\{a,b\}\) is a cut pair and it follows from the paragraph before last that it is inseparable.

\(\rule{.55em}{.55em}\)

The next lemma will allow us to show that limit sets of pared acylindrical submanifolds are inseparable.

When we say that two essential annuli are isotopic we require the isotopy to only go through essential annuli.

Proof. If there is such an essential annulus \(E\) then it has a lift \(\widetilde{E}\) whose endpoints separate \(a\) and \(b\). This is a direct consequence of the following claim which will also be used to prove the converse.

Let \(E\) be an essential annulus, then \(E\) intersects the projection in \(M\) of any arc in \(\mathbb{H}^3\) joining \(a\) to \(b\) if and only if there is a component \(\widetilde{E}\subset\mathbb{H}^3\cup\Omega_K\approx\widetilde{M\setminus P}\) of the preimage of \(E\) such that the closure of \(\partial\widetilde{E} =\widetilde{E}\cap\Omega_K\) in \(\widehat{\mathbb{C}}\) separates \(a\) and \(b\).

Proof. Let \(E\subset M\setminus P\) be an essential annulus. If one boundary curve of \(\partial E\) was homotopic into \(P\), then \(\rho(\pi_1(E))\) would define an accidental parabolic, hence a cut point of \(\Lambda_K\) according to Proposition 48, a contradiction. Therefore its core defines a non-trivial conjugacy class of a loxodromic element \(g\in K\). Let \(\widetilde{E}\) be the lift in \(\mathbb{H}^3\cup\Omega_K\) of \(E\) that is invariant under \(g\).

The closure of \(\widetilde{E}\cap \Omega_K\) is a Jordan curve in \(\widehat{\mathbb{C}}\) that bounds the disc \(\widetilde{E}\cap\mathbb{H}^3\). Let \(\widetilde{\mathcal{E}}\) denote the set of the closures of the lifts of \(E\) and \({\mathcal{C}}\) the collection of their Jordan boundary curves on \(\widehat{\mathbb{C}}\). Note that the elements of \(\widetilde{\mathcal{E}}\) are pairwise disjoint. With the claim in mind, it is also convenient to consider a small toric neighborhood \(T\) of \(E\) whose universal cover \(\widetilde{T}\) may be identified with \(\mathbb{R}\times [-1,1]\times [-1,1]\) and let \(\widetilde{\mathcal{T}}\) be the collection of the closures of its lifts. Its elements are pairwise disjoint as well and their traces on \(\widehat{\mathbb{C}}\) correspond to \({\mathcal{C}}\).

We claim that \(\widetilde{\mathcal{T}}\) is a null sequence when we consider the hyperbolic space as the unit Euclidean ball, i.e., for any \(\delta>0\), there are only finitely many components of diameter at least \(\delta\). To see this, let us assume there exists a sequence of distinct translates \((g_k(\widetilde{T}))_k\) that has diameter bounded from below. Note that as the stabilizer of \(\widetilde{T}\) is cyclic (hence quasiconvex) and malnormal, the set of its fixed points \(c\cap \Lambda_K\), \(c\in{\mathcal{C}}\), is a null-sequence according to Lemma 5 (3). Therefore, we may assume that the set of fixed points \(g_k(Fix(g))\) is convergent towards a point \(x\in \Lambda_K\). Let \(p\in\widehat{\mathbb{C}}\) be disjoint from \(x\) and all elements of \(\widetilde{\mathcal{T}}\) and let us consider the hyperbolic space in the upper half-space model \(\mathbb{C}\times \mathbb{R}_+\) by sending the point \(p\) at infinity. We may assume that the fixed points of \(g\) are \(\{(-1,0),(1,0)\}\). Let \(\gamma\subset\mathbb{H}^3\) be the hyperbolic geodesic joining both points. For each \(k\), consider the hyperbolic isometry \(h_k\) that fixes the point \(p\) and that maps back \(g_k(\gamma)\) to \(\gamma\). As \(h_k g_k\) maps \(\gamma\) to itself and the action of \(g\) on \(\gamma\) is cocompact, we may precompose each \(h_k g_k\) by a suitable power of \(g\) so that \(\{h_kg_k\}\) is relatively compact in \(\mathrm{PSL}(2,\mathbb{C})\). Therefore, the Euclidean diameters of \(h_kg_k(\widetilde{T})\) are uniformly bounded in \(\mathbb{C}\times\mathbb{R}_+\). As each \(h_k\) is affine, it follows that the Euclidean diameter of \(g_k(\widetilde{T})\) is of order the Euclidean diameter of \(g_k(Fix(g))\), hence tends to \(0\) in \(\mathbb{C}\times\mathbb{R}_+\). This shows that \(\widetilde{\mathcal{T}}\) is a null sequence.

Let us now establish the claim. If some \(c\in{\mathcal{C}}\) separates \(\{a,b\}\) in \(\widehat{\mathbb{C}}\), then any curve in \(\mathbb{H}^3\) joining them will intersect \(\widetilde{E}\) by the Jordan theorem. Conversely, assume that \(\{a,b\}\) is not separated by any element of \({\mathcal{C}}\).

We will construct inductively an arc joining \(a\) and \(b\) that is disjoint from the preimage of \(E\). Let us first enumerate \(\widetilde{\mathcal{T}}=\{\widetilde{T}_j\}\) so that their diameters are decreasing with respect to the Euclidean metric when we identify \(\mathbb{H}^3\) with the unit ball. Let \(\kappa_1:[0,1]\to \mathbb{B}^3\) be an arc joining \(a\) and \(b\) that is properly embedded in the unit ball and assume that \(\kappa_j\) has been constructed with the same properties. If \(\kappa_j\cap \widetilde{T}_j=\emptyset\), then set \(\kappa_{j+1}=\kappa_j\). Otherwise, let \(s_j,t_j\) be the first and last point of \(\kappa_j\cap\widetilde{T}_j\). Since \(\{a,b\}\) is not separated by any element of \({\mathcal{C}}\), \(s_j\) and \(t_j\) lie on the same component of \(\partial \widetilde{T}_j\cap\mathbb{H}^3\). The next arc \(\kappa_{j+1}\) is obtained by replacing in \(\kappa_j\) the arc joining \(s_j\) and \(t_j\) with an arc in \(\partial\widetilde{T}_j\cap\mathbb{H}^3\) with the same endpoints. Since \(\widetilde{\mathcal{T}}\) is a null sequence of pairwise disjoint sets each of which contains a point in the unit sphere, for any interval \([x,y]\subset (0,1)\), \(\kappa_1([x,y])\) intersects finitely many sets of \(\widetilde{\mathcal{T}}\). Thus, for any interval \([x,y]\subset (0,1)\), there is \(J\) such that \(\kappa_1([x,y])\cap T_j = \emptyset\) for \(j\geq J\), and so \(\kappa_{j|[x,y]}=\kappa_{J|[x,y]}\) for any \(j\geq J\). It follows that the sequence \((\kappa_j)_j\) converges to a properly embedded arc \(\kappa:[0,1]\to\mathbb{B}^3\) that joins \(a\) and \(b\). By construction, \(\kappa\) is disjoint from the preimage of \(E\). \(\rule{.55em}{.55em}\)

Let \(\{x,y\}\) be a cut pair separating \(a\) and \(b\). By the separation theorem, there is a Jordan curve \(c\subset\widehat{\mathbb{C}}\) separating \(a\) and \(b\) such that \(c\cap\Lambda_K=\{x,y\}\). Let \(k\) and \(k'\) be the components of \(c\setminus\{x,y\}\) and denote by \({\mathcal{O}}\) and \({\mathcal{O}}'\) the component of \(\Omega_K\) containing \(k\) and \(k'\) respectively. Assume that \(k\) and \(k'\) are geodesic with respect to the Poincaré metrics.

If \(\{x,y\}\) is an inseparable cut pair, then, by Lemma 12, \(\{x,y\}\) is the limit set of a conjugate of \(\rho(\pi_1(W))\) for a solid torus component \(W\) of \(M\setminus B\). It follows that \(k\) and \(k'\) are invariant under the action of this conjugate and so they project to simple closed curves that bound an essential annulus. The conclusion follows from the claim above.

Otherwise, by [29] and Lemma 11, \(X=\overline{\mathcal{O}}\cap\overline{{\mathcal{O}}'}\) is the limit set of a conjugate of \(\rho(\pi_1(W))\) where \(W\subset M\) is an essential \(I\)-bundle.

If \(\{a,b\}\subset X\), let \((a,b)\subset\mathbb{H}^3\) be the geodesic joining \(a\) and \(b\) and let \(l\) be its projection to the base surface \(F\) of \(W\) as defined in the proof of Lemma 12. Since \(\{a,b\}\) is not an inseparable cut pair, it follows from Lemma 12 that \(l\) is not peripheral. In particular, there is a simple closed curve \(\gamma\subset F\) that crosses \(l\). The \(I\)-bundle over \(\gamma\) provides us with the desired annulus.

If \(\{a,b\}\not\subset X\), say \(a\not\in X\), let \({\mathcal{H}}\) and \({\mathcal{H}}'\) be the convex hulls of \(X\) in \({\mathcal{O}}\) and \({\mathcal{O}}'\) respectively, equipped with their Poincaré metrics and observe that \(c\subset ({\mathcal{H}}\cup{\mathcal{H}}'\cup X)\). The point \(a\) lies in a component \({\mathcal{A}}\) of \(\widehat{\mathbb{C}}\setminus({\mathcal{H}}\cup{\mathcal{H}}'\cup X)\) and the frontier of \({\mathcal{A}}\) is a Jordan curve \(c'\) that separates \(a\) and \(b\) since either \(c'\) coincides with \(c\) or it separates \(a\) from \(c\) as \(c\subset ({\mathcal{H}}\cup{\mathcal{H}}'\cup X)\), hence separates \(a\) from \(b\) as well. By [29], the projection of \(c'\setminus X\) bounds an annulus and the conclusion follows from the claim above. \(\rule{.55em}{.55em}\)

Proof.

Let \(W\) be a component of \(M\setminus B\), let \(\widetilde{W}\subset\widetilde{M}\) be a connected component of its preimage and denote by \(\Lambda_{\widetilde{W}}\subset\widehat{\mathbb{C}}\) the ideal boundary of \(\widetilde{W}\cap\mathrm{Hull}(\Lambda_K)\). As explained earlier \(\widetilde{W}\cap \mathrm{Hull}(\Lambda_K)\) is quasiconvex, hence \(\Lambda_{\widetilde{W}}\) is the limit set of a conjugate of \(\rho(\pi_1(W))\).

Given two points \(a,b\in \Lambda_{\widetilde{W}}\), we can use the same construction as in Lemma 13 to build an arc \(l\subset \widetilde{W}\) joining \(a\) and \(b\). As mentioned in the proof of Lemma 8, any essential annulus can be made disjoint from \(B\) by an isotopy (see [88]). When \(W\) is acylindrical, it follows then from Lemma 13 that \(\Lambda_{\widetilde{W}}\) is inseparable.

On the other hand, let \(X\subset\Lambda_K\) be an inseparable set and let \(\widetilde{B}\subset\mathbb{H}^3\cup\Omega_K\) be the preimage of \(B\). Replacing \(E\) with \(B\) in the claim inside the proof of Lemma 13, we may conclude that any two points in \(X\) are joined by a line that is disjoint from \(\widetilde{B}\). Thus there is a component \(\widetilde{W}\) of \((\mathbb{H}^3\cup\Omega_K)\setminus\widetilde{B}\) such that \(\widetilde{W}\) contains a line joining \((a,b)\) for any pair \((a,b)\in X^2\). It follows that \(X\) is a subset of the ideal boundary \(\Lambda_{\widetilde{W}}\) of \(\widetilde{W}\). We have established at the beginning of the proof that \(\Lambda_{\widetilde{W}}\) is the limit set of a conjugate of \(\rho(\pi_1(W))\). If \(X\) has at least \(3\) points, \(W\) can not be a solid or thickened torus and by Lemma 11 it can not be an essential \(I\)-bundle when \(X\) is also inseparable.

We have shown that a subset \(X\) of \(\Lambda_K\) with at least \(3\) points is inseparable if and only if there is a pared acylindrical component \(W\) of \(M\setminus B\) such that \(X\) lies in the limit set of a conjugate of \(\rho(\pi_1(W))\). It follows that the maximal inseparable sets with at least \(3\) points are exactly the limit sets of conjugates of fundamental groups of pared acylindrical components of \(M\setminus B\).

\(\rule{.55em}{.55em}\)

This concludes the proof of Proposition 49.

Let us now recall the definition of \({\mathcal{T}}_{\Lambda_K}\). Let \(K\) be a torsion free geometrically finite Kleinian group uniformizing a pared manifold \((M,P)\) and consider the set \(V\) of cut points, inseparable cut pairs, necklaces and maximal inseparable sets with at least 3 points of \(\Lambda_K\). By [92] and Proposition 49, \(V\) is countable. Define a graph \({\mathcal{T}}_{\Lambda_K}\) with vertex set \(V\) by putting an edge between two vertices \(v_1,v_2\) if \(v_1\subset v_2\) as subsets of \(\Lambda_K\). By Lemma 8, Lemma 9 and Proposition 49, \({\mathcal{T}}_{\Lambda_K}\) is the tree dual to the balanced annulus decomposition of the pared manifold \((M,P)\) uniformized by \(K\). Thus we have:

From now on, we will call \({\mathcal{T}}_{\Lambda_K}\) the JSJ tree of \(K\). Let us notice that any homeomorphism of \(\Lambda_K\) induces an isometry on \({\mathcal{T}}_{\Lambda_K}\).

4.4.4 Quasi-isometries and JSJ splittings↩︎

Let \(G\) be a finitely generated group quasi-isometric to a one-ended Kleinian group \(K\). By Remark 17, we may assume that \(K\) is torsion free, geometrically finite and minimally parabolic. To obtain the splitting from Theorem 45, we will prove that the Bowditch boundary of \(G\) is homeomorphic to \(\Lambda_K\). Thus we get a simplicial action of \(G\) on the JSJ tree \({\mathcal{T}}_{\Lambda_K}\) and a splitting of \(G\). We will then deduce properties (1) to (7) from the results of §4.4.2, §4.4.3 and the quasi-isometric invariance of the JSJ splitting.

Since \(K\) is a geometrically finite Kleinian group, it is hyperbolic relative to its parabolic subgroups. Since it is minimally parabolic, they are isomorphic to \(\mathbb{Z}^2\). By [97], \(G\) is hyperbolic relative to a collection \(\mathbb{P}\) of subgroups each of which is quasi-isometric to \(\mathbb{Z}^2\). It follows then from Theorem 7 that the peripheral subgroups of \(G\) are virtually \(\mathbb{Z}^2\). Furthermore, by results of Osin and Drutu-Sapir the peripheral subgroups are quasi-isometrically embedded [71] and, by [97], the quasi-isometry maps each coset of a peripheral subgroup in a bounded neighborhood of a coset of a parabolic subgroup and conversely. Thus we get:

By Theorem 34, the quasi-isometry in Lemma 15 can be extended to a quasi-isometry between cusped spaces \(X_G\) and \(X_K\) which, by Proposition 35, induces a quasi-isometry between \(X_G\) and the convex hull of \(\Lambda_K\). Such a quasi-isometry extends to a homeomorphism from the Bowditch boundary \(\partial_\mathbb{P}G\) to the limit set \(\Lambda_K\) of \(K\). Thus we have:

The action of \(G\) on its Bowditch boundary induces a simplicial action on the JSJ tree \({\mathcal{T}}_{\Lambda_K}\). Notice that by definition a vertex group of the JSJ splitting of \(G\) is the stabilizer of a cut point, an inseparable cut pair, a necklace or a maximal inseparable set in \(\partial_\mathbb{P}G\). Recall that two vertices \(v_1,v_2\) are adjacent if and only if \(v_1\subset v_2\) as subsets of \(\Lambda_K\) (or the opposite). This can only happen if \(v_1\) is a cut point or cut pair and \(v_2\) is a necklace or maximal inseparable set with at least 3 points. It follows that the action of \(G\) on \({\mathcal{T}}_{\Lambda_K}\) has no edge inversion and that an edge group stabilizes two subsets \(X,Y\) of \(\Lambda_K\) where \(Y\) is a cut point or cut pair and \(X\) is a necklace or maximal inseparable set with at least 3 points containing \(Y\). The action of \(G\) on \({\mathcal{T}}_{\Lambda_K}\) induces a splitting, which we call the JSJ splitting of \(G\). This in turn fits in the more general setting of JSJ decompositions of finitely generated groups established in [89]. The last ingredient we need to prove Theorem 45 is the quasi-isometric invariance of the JSJ splitting.

Using Proposition 36, it is easy to show that \(G_X\) is quasi-isometric to \(K_X\), once we have established that \(X\) is their limit set.

Proof. If the JSJ decomposition is trivial, then there is nothing to prove. So let us assume that \(X\) is a proper subset of \(\partial_\mathbb{P}G\). By Proposition 49, we know that \(X\) corresponds to the limit set of \(K_X\), so that it contains a dense collection of cut points and/or inseparable cut pairs. But, for each cut point or inseparable cut pair \(Y\subset X\), it follows from the definition of \({\mathcal{T}}_{\Lambda_K}\) that there is an edge stabilizer in \(G\) which stabilizes both \(X\) and \(Y\). Since \(G\) is one-ended, edge groups are infinite, cf. [2]. It follows that \(Y\subset\Lambda_{G_X}\), and by density of such sets in \(X\), that \(X\subset \Lambda_{G_X}\). Since \(X\) is stabilized by \(G_X\), we finally get \(X = \Lambda_{G_X}\). \(\rule{.55em}{.55em}\)

. By Lemmas 15 and 16, there is a quasi-isometry \(\varphi:G\to K\) that preserves the parabolic structure and that defines a homeomorphism \(\partial\varphi:\partial_\mathbb{P}G\to \Lambda_K\). By

[98] (see also [99] and [75]), \(G_X\) and \(K_X\) are relatively quasiconvex. It follows from Fact 51 and Proposition 36 that \(\varphi(G_X)\) is at bounded distance from \(K_X\) and Fact 28 ensures that \(G_X\) and \(K_X\) are quasi-isometric. It remains to prove that the quasi-isometry preserves the parings.

Proposition 36 already ensures that the parabolic structure is preserved, so we only need to consider the stabilizers of inseparable cut pairs. Let \(P\in \mathbb{P}_X\) be the stabilizer in \(G_X\) of an inseparable cut pair \(C\subset X\). We know that \(P\) is virtually cyclic and \(P\subset G\) is a quasigeodesic with end points \(C\). Then \(\varphi(P)\) is a quasigeodesic with endpoints \(\partial\varphi(C)\). Since \(\partial\varphi(C)\) is an inseparable cut pair, there are \(k\in K\) and \(Q\in \mathbb{Q}_X\) such that \(kQk^{-1}\) stabilizes \(\partial\varphi(C)\) and \(\partial\varphi(X)\). Since \(Q\) is cyclic, \(kQ\) is a quasigeodesic with endpoints \(\partial\varphi(C)\). Now \(\varphi(P)\) and \(kQ\) are quasigeodesics with the same endpoints, hence their distance is bounded.

This proves that \((G_X,\mathbb{P}_X)\) and \((K_X, \mathbb{Q}_X)\) are quasi-isometric as pared groups. \(\rule{.55em}{.55em}\)

We are now ready to prove Theorem 45.

. Let \(G\) be a finitely generated one-ended group quasi-isometric to a torsion free geometrically finite minimally parabolic Kleinian group \(K\), cf. Remark 17. By Lemma 15, \(G\) is hyperbolic relative to some collection of subgroups \(\mathbb{P}\) made up of virtually Abelian rank \(2\) subgroups. Lemma 16 provides an action of \(G\) on the JSJ tree \({\mathcal{T}}_{\Lambda_K}\). We will show that the induced JSJ splitting has Properties (1) to (7)

, starting with the easiest properties.

Since \(G\) is hyperbolic relative to subgroups virtually isomorphic to \(\mathbb{Z}^2\) (Lemma 15), virtual Abelian subgroups are finite, virtually cyclic or virtually \(\mathbb{Z}^2\). Since \(G\) is one-ended, edge and vertex groups are infinite, cf. [2]. Property (3) follows.

Let us establish Property (1). As previously mentioned an edge group stabilizes in \(\partial_\mathbb{P}G\), two subsets \(X,Y\) where \(Y\) is a cut point or cut pair and \(X\) is a necklace or maximal inseparable set with at least 3 points containing \(Y\). We treat two cases depending on whether \(Y\) is an inseparable cut pair or a cut point. If an infinite subgroup of \(G\) stabilizes a pair of points \(\{a,b\}\), then it is virtually cyclic since it is elementary and 2-ended. Thus, this edge group is virtually cyclic, i.e. Property (1) holds in that case. If an infinite subgroup of \(G\) stabilizes a single point, it is a subgroup of a peripheral subgroup of \(G\) which is virtually \(\mathbb{Z}^2\) by Lemma 15. Hence, if an edge group stabilizes a cut point \(Y=\{a\}\) and a necklace or a maximal inseparable set \(X\) then it also stabilizes the connected component of \(\partial_\mathbb{P}G\setminus\{a\}\) containing \(X\). Property (1) follows in that case from Lemma 7.

Let us prove Property (2). Let \(v\) be a vertex with (infinite) virtually Abelian group \(G_v\). By Property (3) established above, it contains a finite index subgroup isomorphic to \(\mathbb{Z}\) or \(\mathbb{Z}^2\). Any edge group incident to a virtually cyclic vertex group is commensurable to that vertex group since both are 2-ended; Property (2) follows in that case. Lemma 7 combined with the argument in the previous paragraph also establishes Property (2) when \(G_v\) is commensurable to \(\mathbb{Z}^2\).

A group of isometries of a Gromov hyperbolic space is virtually Abelian only if it is elementary, i.e., its limit set is made up of at most two points. A consequence of Fact 51 is that virtual Abelian vertex groups are precisely the stabilizers of cut points and inseparable cut pairs. Property (4) follows then from the definition of the JSJ tree.

Property (5) follows from Proposition 50 and the fact that the JSJ tree is dual to the balanced annulus decomposition by Corollary 4.

Finally by Lemma 15, the maximal parabolic subgroups of \(K\) are mapped by the quasi-isometry at bounded Hausdorff distance from those of \(G\), so both \(G\) and \(K\) have the same parabolic points. Since (6) and (7) hold for \(K\), by Corollary 4, they also hold for \(G\). \(\rule{.55em}{.55em}\)

5 Prescription of subgroups up to finite index↩︎

We show how to construct finite index subgroups of JSJ decompositions with prescribed vertex and edge groups.

5.1 Finite index subgroups with prescribed peripheral subgroups↩︎

Let \(G\) be a group. A subgroup \(H<G\) is separable if, for any \(g\in G\setminus H\), there exists a finite index subgroup \(G'<G\) which contains \(H\) but not \(g\). The group \(G\) is residually finite if \(\{1\}\) is separable; in other words, for any \(g\ne 1\), there exists a finite index subgroup \(G'<G\) disjoint from \(g\). Equivalently, for any \(g\ne 1\), there exists a normal finite index subgroup \(G'<G\) disjoint from \(g\). A group is LERF (Locally Extended Residually Finite) if any finitely generated subgroup is separable. Two groups \(G\) and \(Q\) are commensurable when \(G\) has a finite index subgroup isomorphic to a finite index subgroup of \(Q\). Notice that if \(Q\) is residually finite or virtually torsion free, then \(G\) is as well.

Let us explain the analogy with three-manifolds. A Dehn filling of a compact three-manifold \(M\) consists in gluing a solid torus along a toroidal boundary component of \(M\). On the level of fundamental groups, if \(P<\pi_1(M)\) is the fundamental group of the toroidal boundary component, a Dehn filling yields a subgroup \(P^c <P\) in each \(P\) such that the fundamental group of the filled manifold is isomorphic to \(\pi_1(M)/ \ll P^c,\;P\in\mathbb{P}\gg\). Note that in this topological setting, the subgroup \(P^c\) is cyclic with infinite index in \(P\).

We will use this property to find finite index pared subgroups with prescribed parings through the following claim:

Proof. By construction \(\overline{P_j}=P_j/P_j^c\) is finite. By property \((\ast)\), \(\overline{G}\) is residually finite and there is a morphism \(f:\overline{G}\to Q\) to a finite group \(Q\) such that \(1\notin f(\overline{P_j}\setminus\{1\})\) for all \(j\). Let \(H\) be the kernel of \(G\to \overline{G}\to Q\). Then \(H\) is a normal finite index subgroup of \(G\) such that, for any \(g \in G\) and any \(j\in\{1,\ldots, n\}\), \(H\)\(\cap g P_{j} g^{-1} = g P_{j}^c g^{-1}\). We obtain a paring from Fact 33. \(\rule{.55em}{.55em}\)

Our chief concern for the deep residually finite Dehn filling property (\(\ast\)) is to Kleinian groups, but this notion could be of interest for other classes. It is inspired by the work of Wise [31].

Proof. We start by observing that, by [31], \(K\), and hence \(G\), is virtually compact special. Then using a generalization of the malnormal special quotient theorem [31], see also [100], we establish the existence of \(P_j^\circ <P_j\) such that, whenever \(P^c_j < P^\circ_j\) is a normal finite index subgroup of \(P_j\) for each \(j\), the quotient \(\overline{G}=G/\ll P^c_j,\;P_j\in\mathbb{P}\gg\) is also virtually compact special. The conclusion that \(\overline{G}=G/\ll P^c_j,\;P_j\in\mathbb{P}\gg\) is residually finite follows from the residual finiteness of compact special groups, [101] and [102] and the fact that \(P_j^\circ\) can be chosen so that \(\overline{P}_j=P_j/P^c_j\) embeds in \(\overline{G}\) follows from [103]. \(\rule{.55em}{.55em}\)

5.2 Graph of groups structure with prescribed vertex groups↩︎

We now use the previous results to “choose” the vertex and edge groups in a graph of groups.

1.  *To a vertex of type $A$ corresponds a group $G_v$ which admits
    a paring $(G_v,\mathbb{P}_v)$ containing the adjacent edge
    groups and which has the deep residually finite Dehn fillings
    property ($\ast$).*

2.  *To a vertex of type $B$ corresponds a group $G_v$ such that any
    finite index subgroup of an adjacent edge group is separable in
    $G_v$.*

Proof. For each vertex subgroup \(G_v\), we will find a finite index subgroup \(G_v'\) of \(H_v\), normal in \(G_v\), so that if \(e\in E\) is an edge, then \(j_e^{-1}(G'_{t(e)})= j_{\bar e}^{-1}(G'_{t(\bar e)})\). This will allow us to construct a quotient of \(G\) in order to form a finite index subgroup of \(G\) that combines these subgroups together.

Considering instead their normal core, we may assume that the subgroups \(H_v\) and \(H_e\) are normal (and of finite index) in \(G_v\) and \(G_e\) respectively, and we may also suppose that \(H_e=H_{\bar e}\).

For \(v\in A\), property (\(\ast\)) provides us for each \(P\in\mathbb{P}_v\) a finite index subgroup \(P^\circ <P\) which fixes us the necessary deepness to perform controlled Dehn fillings. We may assume that \(P^\circ\) is also normal in \(P\). For \(P= j_e (G_e)\), \(v=t(e)\), we will also write \(P_e= P\) and \(P_{e}^\circ = P^\circ\). It is important here to remember that the edges are oriented and that the treatments for \(e\) and \(\bar e\) are independent since they depend on their terminal points. In particular \(P_{\bar e}^\circ\) might not be defined, let alone equal to \(P_e^\circ\).

Let \(e\in E\); by condition (ii) of the statement, either a single extremity is in \(A\), and then, we may assume that it is \(t(e)\), or both extremities are in \(A\).

In the latter case, set \(K_e= K_{\bar e}= H_e\cap j_e^{-1}(H_{t(e)}\cap P_{e}^\circ) \cap j_{\bar e}^{-1}(H_{t(\bar e)}\cap P_{\bar e}^\circ)\). In the former case, set \(K_e= H_e\cap j_e^{-1}(H_{t(e)}\cap P_{e}^\circ)\). Note that, in both cases, \(K_e\) is a finite index subgroup of \(j_e^{-1}(P_e^\circ)\), contained in \(H_e\) and normal in \(G_e\) (as intersection of normal subgroups).

Fix \(v\in B\) and let \(e\) be an incident edge so that \(t(e)=v\). Let \(I\) be the (finite) collection of edges \(f\in t^{-1}(v)\) incident to \(v\) for which \(j_f(G_f)\) is commensurable to \(j_e(G_e)\). Let \(L_e =L_I = H_v \cap ( \cap_{f\in I} j_f(K_f))\); this is a finite index subgroup of \(j_e(G_e)\). Therefore, since \(v\) is in \(B\), there is a normal finite index subgroup \(G_{v,e}=G_{v,I} <H_v\) in \(G_v\) such that \(G_{v,e}\cap j_e(K_e)<L_e\) is a normal finite index subgroup of \(j_e(G_e)\). Note that, under the condition (i) (b’), if \(G_e\) is virtually cyclic, since, by assumption, \(G_{v,e}\cap j_e(K_e)\) is separable in \(H_v\), up to taking a finite index subgroup, we may assume that \(G_{v,e}\cap j_e(K_e)\) is generated by a primitive element within \(G_{v,e}\).

Now, we define \(G_v'= \cap_{t(e)=v} G_{v,e}\). This is a normal finite index subgroup of \(G_v\) such that, for any edge \(e\) with \(t(e)=v\), \(G_e' = j_e^{-1}(G'_v)\) is a normal finite index subgroup of \(G_e\) contained in \(j_{\bar e}^{-1} (P_{t(\bar e)}^\circ)\). We let \(G_{\bar e}'=G_e'\). It follows from the previous paragraph that if two edges \(e_1\) and \(e_2\) have commensurable groups \(G_{e_1}\) and \(G_{e_2}\) in \(G_v\), then \(G_{e_1}'\) and \(G_{e_2}'\) now coincide in \(G_v'\) and if \(j_e(G_e')\) is cyclic, then it is generated by a primitive element.

If \(e\) is an edge for which both extremities are in \(A\), we set \(G_e'=K_e\).

Let us now consider \(v\in A\). Let \(e\) be an edge with \(t(e)=v\) and let \(P_e^c= j_e(G_e')\). For the peripheral subgroups \(P\) in \(\mathbb{P}_v\) which are not edges, we consider any normal finite index subgroup \(P^c< P^\circ\) of \(P\). The deep residually finite Dehn filling property provides us through Claim 53 with a normal finite index subgroup \(K_v{\subset G_v}\) such that \(K_v\cap P= P^c\) for all \(P\in\mathbb{P}_v\). Set \(G_v'=K_v\cap H_v\) that is by construction a finite index normal subgroup of \(G_v\). We note that since \(j_e(K_e)\subset H_{t(e)}\) for every edge \(e\in E\), we obtain for each \(v\in A\) and \(e\in t^{-1}(v)\), \(G_v'\cap j_e(G_e)= P_e^c= j_e(G_e')\).

By construction, \(j_e^{-1}(G'_{t(e)})= G_e'= j_{\bar e}^{-1}(G'_{t(\bar e)})\) holds for each edge \(e\). We may consider the graph of groups \[\overline{{\mathcal{G}}}=(\Gamma,\{\overline{G}_v\},\{\overline{G}_e\}, \overline{G}_e\hookrightarrow \overline{G}_{t(e)})\] where \(\overline{G}_v= G_v/G_v'\) and \(\overline{G}_e= G_e/ G_e'\). Let \(\overline{G}\) be the fundamental group of \(\overline{{\mathcal{G}}}\), which is a finite graph of finite groups, hence is virtually free and residually finite.

The canonical projections \(G_v\to\overline{G}_v\) define a projection \(p:G\to\overline{G}\). Since \(\overline{G}\) is residually finite, there is a morphism \(\varphi:\overline{G}\to F\) to a finite group \(F\) that maps a set of transversals for each \(G'_v<G_v\) to distinct elements. Let \(G' =\ker (\varphi\circ p)\) be the kernel of \((G\to\overline{G}\to F)\). Then \(G'\cap G_v=G'_v\) for any vertex \(v\). The action of \(G'\) on the Bass-Serre tree of \(G\) defines a finite graph of groups structure for which vertex groups are conjugate to some subgroup \(G'_v\) and edge groups are conjugate to some subgroup \(G_e'\). \(\rule{.55em}{.55em}\)

We draw the following application to three-manifolds:

Proof. Taking a degree two cover, we may assume that \(M\) is orientable. Let us introduce the GH-decomposition \(T_{GH}\subset T_B\), which is the minimal union of components of \(T_B\) such that the compactification of a component of \(M\setminus T_{GH}\) is either a graph manifold or atoroidal (and hence hyperbolic). This decomposition provides us with a graph of groups structure (which may be trivial) such that each vertex group is the fundamental group of a graph manifold or a hyperbolic manifold. By [12] and [104] (see also [14]), if \(N_i\) is graph manifold appearing in this decomposition then \(N_i\) has a finite cover \(M_i\) such that all the pieces in the characteristic torus decomposition of \(M_i\) are products. By [105] and Theorem 54, the hypotheses of Proposition 55 are fulfilled, and it produces a normal finite index subgroup \(G'\) so that each Seifert piece corresponds to a product \(\mathbb{Z}\times \pi_1(S)\) where \(S\) is an orientable surface. The corresponding cover satisfies the conclusions of the proposition.\(\rule{.55em}{.55em}\)

6 Quasi-isometric rigidity of three-manifolds groups↩︎

In this section, we gather our previous results to establish Theorems 2, 4 and 3, starting with the latter since it is used in the two other proofs.

6.1 Quotients by finite subgroups vs finite index subgroups↩︎

Our main use for separability is the following proposition, see [16] for a proof.

Our goal here is to present some results to conclude that groups quasi-isometric to three-manifold groups are residually finite, cf. Theorem 3.

Proof. We will produce a morphism \(q:G\to\overline{G}\) onto a virtually free group which is injective on \(\ker p\), and use the separability of free groups to conclude. The construction of the projection \(q\) will result from taking compatible quotients of the vertex groups of \(G\) onto finite subgroups.

The action of \(Q\) on its Bass-Serre tree yields through \(p\) an action of \(G\), inducing a graph of groups structure \[(\Gamma,\{G_v=p^{-1}(Q_v)\},\{G_e=p^{-1}(Q_e)\}, j_e:p^{-1}(Q_e)\hookrightarrow p^{-1}(Q_{t(e)}))\] for \(G\) such that \(i_e\circ p= p\circ j_e\) for every edge (this can be deduced for example from [81]).

By assumption each vertex group \(G_v=p^{-1}(Q_v)\) is residually finite, hence there is a normal finite index subgroup \(G'_v\) of \(G_v\) such that \(G_v'\cap \ker p = \{1\}\).

For any edge \(e\), set \(G_e'= j_e^{-1}(G_{t(e)}')\cap j_{\bar e}^{-1}(G_{t(\bar e)}')\). By construction, the restriction \(p:G_e'\to Q_e\) is injective and the group \(Q_e'= p(G_e')\) is a normal finite index subgroup of \(p(G_e)=Q_e\). By assumption, \(Q_e'\) is separable and by Proposition 57, there is a normal finite index subgroup \(N_e\) in \(Q\) such that \(N_e\cap Q_e\subset Q'_e\).

Let \(Q'= \cap_e N_e\), which has finite index in \(Q\) since \(N_e\) has finite index in \(Q\) for any edge \(e\) and \(\Gamma\) is finite. For each vertex \(v\), set \(H_v= p^{-1}(Q')\cap G_v'\). This subgroup \(H_v\) is normal and has finite index in \(G_v\) since the same holds for \(G_v'\) in \(G_v\) and for \(Q'\) in \(Q\); furthermore, we have \(H_v\cap \ker p\subset G_v'\cap \ker p = \{1\}\).

Let us fix an edge \(e\). Since \(p\) is injective on \(j_e^{-1}(G'_{t(e)})\cup j_{\bar e}^{-1}(G_{t(\bar e)}')\) and \(Q'\cap p(G_e)\subset Q_e'\), we have \[j_e^{-1}(H_{t(e)}) = (p\circ j_e)^{-1}(Q')\cap G_e'= (p\circ j_{\bar e})^{-1}(Q')\cap G_e' =j_{\bar e}^{-1}(H_{t(\bar e)})\] and this subgroup that we name \(H_e\) is normal in \(G_e\).

This implies that we may define a graph of groups \[\overline{{\mathcal{G}}}=(\Gamma,\{\overline{G}_v\},\{\overline{G}_e\}, \overline{G}_e\hookrightarrow \overline{G}_{t(e)})\] where \(\overline{G}_v= G_v/H_v\) and \(\overline{G}_e= G_e/H_e\). Let \(\overline{G}\) be the fundamental group of \(\overline{{\mathcal{G}}}\), which is a finite graph of finite groups, hence \(\overline{G}\) is virtually free [82] and residually finite.

The canonical projections \(G_v\to\overline{G}_v\) define a projection \(q:G\to\overline{G}\) and \(q\) is injective on \(\ker p\). Since \(\overline{G}\) is residually finite, there is a morphism \(r:\overline{G}\to F\) to a finite group which is injective on \(q(\ker p)\). The kernel \(H= \ker (r\circ q)\) is a finite index subgroup of \(G\) which avoids \(\ker p\setminus\{1\}\), so \(H\) embeds in \(Q\) as a finite index subgroup. \(\rule{.55em}{.55em}\)

Let \(G\) be a group. A hierarchy of length \(0\) for \(G\) is the graph of groups made of the single vertex group \(G\). Let \(n>0\), a hierarchy of length at most \(n\) for \(G\) is a non-trivial finite graph of groups structure for \(G\) together with a hierarchy of length at most \(n-1\) for each vertex group. The hierarchy has length \(n\) if it has length at most \(n\) and at least one vertex group has a hierarchy of length \(n-1\).

Let \(G_v\) be a vertex group of the graph of groups structure of \(G\). We say that \(G_v\) is at level \(1\) of the hierarchy. The groups at level \(1\) for the hierarchy of \(G_v\) are at level \(2\) of the hierarchy for \(G\) and so on. The vertex groups with a hierarchy of length \(0\) are called the terminal groups of the hierarchy.

Using Lemma 17 inductively on a complete hierarchy, we get the following:

Proof. From the hierarchy for \(Q\) the morphism \(p\) induces a hierarchy of length \(n\) for \(G\) with the property that if \(G_v\) is a group at level \(k\) of the hierarchy for \(G\) then \(G_v=p^{-1}(Q_v)\) for some group \(Q_v\) at level \(k\) of the hierarchy for \(Q\).

We are going to prove Lemma 18 with a finite recurrence on the level in reverse order. Our induction hypothesis is:

1. Any group \(G_v\) at level \(k\) of the hierarchy for \(G\) is residually finite and commensurable to \(Q_v=p(G_v)\).

\(P(n)\) is satisfied by assumption.

Assume \(P(k)\) holds for a given \(k\) (\(1\leq k\leq n\)) and let us show that \(P(k-1)\) holds. Let \(G_v\) be a group at level \(k-1\) of the hierarchy for \(G\), by construction, \(Q_v=p(G_v)\) is a group at level \(k-1\) of the hierarchy for \(Q\). Since \(P(k)\) holds, by the assumption made on the hierarchy for \(Q\), \(G_v\) and \(Q_v\) satisfy the assumption of Lemma 17. It follows that \(G_v\) is commensurable to \(Q_v=p(G_v)\). Since \(Q\) is residually finite, the same is true for \(Q_v\), hence for \(G_v\). \(\rule{.55em}{.55em}\)

We may now prove Theorem 3.

Proof. Up to taking a finite index subgroup of \(G\), we may assume that \(Q\) is isomorphic to the fundamental group of \(M\) and that \(M\) is orientable. If \(G\) is finite, \(G\) and \(Q\) are commensurable to the trivial group, so we may assume that \(G\) and \(Q\) are infinite.

If \(M\) is a surface, it is easy to find a hierarchy for \(Q\) with trivial terminal groups and finitely generated edge groups (see [82] for example). According to [106], [107], \(Q\) is LERF and the conclusion follows from Lemma 18.

If \(M\) is a compact irreducible three-manifold, we know from Agol [108] that \(M\) is virtually Haken. So, up to taking a finite index subgroup of \(G\), we may assume that \(M\) is Haken.

Let \(T\) be the characteristic torus decomposition of \(M\). If \(T\) is empty, then \(M\) is Seifert or atoroidal (hence hyperbolic by Theorem 14). Fundamental groups of Seifert and hyperbolic manifolds are LERF by [106], [107] for Seifert manifolds and [108] for hyperbolic manifold (see also [31]). The Haken hierarchy [109], [110] in \(M\) provides us with a hierarchy for \(Q\) with trivial terminal groups and finitely generated edge groups. Again we conclude with Lemma 18.

If \(T\neq\emptyset\), which also includes the case of Sol manifolds, then it induces a graph of groups structure \((\Delta,\{Q_v\},\{Q_e\}, Q_e\hookrightarrow Q_{t(e)})\) for \(Q\) with edge groups \(Q_e\) isomorphic to \(\mathbb{Z}^2\) and vertex groups \(G_v\) isomorphic to fundamental groups of Seifert or hyperbolic manifold \(M_v\). By [105], any subgroup of an edge group is separable in \(Q\). By the hyperbolization theorem, every vertex manifold \(M_v\) is geometric, in particular \(Q_v\) is linear and hence residually finite by Malcev’s theorem [102]. We have already proved Theorem 3 for Seifert and hyperbolic manifolds so we know that \(p^{-1}(Q_v)\) is comensurable to \(Q_v\). Thus \(p^{-1}(Q_v)\) is residually finite and the conclusion follows from Lemma 17.

If \(M\) contains an essential sphere, \(Q\) has a graph of groups structure with trivial edge groups and vertex groups isomorphic to fundamental groups of compact irreducible three-manifolds. It follows from the previous paragraphs that Theorem 3 holds for compact irreducible three-manifolds. Since fundamental groups of compact three-manifolds are residually finite [111], the conclusion follows again from Lemma 17. \(\rule{.55em}{.55em}\)

Lemma 18 can also be used to prove a strong version of the Bieberbach theorem where the classical faithfulness assumption is removed. Notice that this generalization is already well known and was used in the proof of the quasi-isometric rigidity of virtually Abelian groups by Cornulier-Tessera-Valette [7].

Proof. Let \(K\triangleleft G\) be the kernel of this action, which is finite since the action is proper. Then \(G/K\) acts properly, faithfully and cocompactly on \(\mathbb{E}^n\), hence by the Bieberbach theorem, it is virtually \(\mathbb{Z}^n\). Thus we have a morphism \(p:G'\to \mathbb{Z}^n\) with finite kernel where \(G'\) is a finite index subgroup of \(G\). For any \(k\geq 1\), \(\mathbb{Z}^k\) is an HNN extension of \(\mathbb{Z}^{k-1}\), i.e., has a graph of groups structure for which the graph has a single vertex and a single edge, and where both vertex and edge groups are \(\mathbb{Z}^{k-1}\). This leads us to a hierarchy for \(\mathbb{Z}^n\) that satisfies the assumptions of Lemma 18. It follows then that \(G'\) and \(\mathbb{Z}^n\) are commensurable and that \(G\) is virtually \(\mathbb{Z}^n\). \(\rule{.55em}{.55em}\)

6.2 Acylindrical hyperbolic manifolds and \(I\)-bundles↩︎

The goal of this section is to establish the quasi-isometric rigidity of non-Abelian vertex groups of a JSJ splitting in the sense of §4.4. More concretely we are going to show the quasi-isometric rigidity of pared fundamental groups of pared \(I\)-bundles and of acylindrical hyperbolic pared manifolds.

Proof. Let \((W,\mathbb{P}_W)=(F\times I,\partial F\times I)\) be a pared \(I\)-bundle whose pared fundamental group \((K,\mathbb{P}_K)\) is quasi-isometric to \((G,\mathbb{P}_G)\). If \(\partial F=\emptyset\), the conclusion follows from Theorem 9. Otherwise \(K\) is a free group and by Theorem 5, \(G\) has a free finite index subgroup \(H\). The induced paring \((H,\mathbb{P}_H)\) given by Fact 33 is thus quasi-isometric to \((W,\mathbb{P}_W)\). Considering a finite volume hyperbolic structure on the interior of \(F\) and applying Proposition 35, we see that \(\partial\mathcal{C}\!\it us\,(K,\mathbb{P}_K)\) is homeomorphic to \(S^1\). By Theorem 34 \(\partial\mathcal{C}\!\it us\,(H,\mathbb{P}_H)\) is also homeomorphic to a circle and by Proposition 37 and [76], \((H,\mathbb{P}_H)\) is the pared fundamental group of a pared \(I\)-bundle. \(\rule{.55em}{.55em}\)

To prove the quasi-isometric rigidity of pared fundamental groups of acylindrical hyperbolic manifolds, we first establish the rigidity of their action on their limit set.

We start with a preliminary rigidity result concerning quasi-Möbius mappings. A Schottky set is the complement of a family of at least three pairwise disjoint open round discs of \(\widehat{\mathbb{C}}\).

The proof of Theorem 59 is essentially the same as Corollary 3.9 in [17].

. Let \(\Lambda_K\) be the limit set of \(K\), since \(K\) is geometrically finite and its convex core has totally geodesic boundary, \(\Lambda_K\) is a Schottky set and has measure \(0\) by [52] and [113]. Let \(F\) be the kernel of the action of \(G\) on \(\Lambda_K\) and \(G'=G/F\). Since \(G\) acts properly discontinuously on distinct triples in \(\Lambda_K\), \(F\) is finite. Let \(G_M\) denote the set of quasi-Möbius selfhomeomorphisms of \(\Lambda_K\). Note that \(G_M\) contains \(K\cup G'\).

According to Theorem 60, the action of \(G_M\) extends to an action of Möbius transformations on \(\widehat{\mathbb{C}}\). It is clearly discrete since any sequence which tends uniformly to the identity will have to eventually stabilize at least three circles, implying that such a sequence is eventually the identity. Since the limit sets of \(K\) and \(G_M\) coincide and since \(K\) is geometrically finite, \(K\) has finite index in \(G_M\), cf. [114]. Since the action of \(G'\) on \(\Lambda_K\) is minimal, \(\Lambda_K\) is the limit set of \(G'\). By assumption this action is geometrically finite, hence again by [114], \(G'\) has finite index in \(G_M\). Thus \(G'\cap K\) has finite index both in \(G'\) and \(K\). By Theorem 3, \(G'\cap K\) has a finite index subgroup that embeds in \(G\) and the conclusion follows. \(\rule{.55em}{.55em}\)

We generalize Theorem 59 as follows:

Notice that the closed and finite volume cases follow from [41] and [42] respectively and that the case of an acylindrical pared free group has been established by Otal [76].

. Let \((G,\mathbb{P}_G)\) be a pared group quasi-isometric to the pared fundamental group of an acylindrical pared hyperbolic manifold \((M,P)\). By Theorem 15, there is a geometrically finite Kleinian group \(K\) whose convex core is homeomorphic to \(M\setminus P\) and has totally geodesic boundary. As previously mentioned, when \(\mathbb{H}^3/K\) has finite volume, the conclusion follows from [41] and [42]. So let us assume that the volume of \(\mathbb{H}^3/K\) is infinite. Let \(\mathbb{P}_K\) be the paring of \(K\) given by its parabolic subgroups. With respect to the quasi-isometry between \((G,\mathbb{P}_G)\) and \((K,\mathbb{P}_K\)), the image of any coset of an element of \(\mathbb{P}_G\) is at bounded Hausdorff distance from a coset of an element of \(\mathbb{P}_K\) and conversely. By Theorem 34, the quasi-isometry between \((G,\mathbb{P}_G)\) and \((K,\mathbb{P}_K)\) extends to a quasi-isometry between cusped spaces \(\mathcal{C}\!\it us\,(G,\mathbb{P}_G)\) and \(\mathcal{C}\!\it us\,(K,\mathbb{P}_K)\). It follows that \(\mathcal{C}\!\it us\,(G,\mathbb{P}_G)\) is hyperbolic and that \(G\) is hyperbolic relative to \(\mathbb{P}_G\). By Proposition 35 the quasi-isometry between cusped spaces \(\mathcal{C}\!\it us\,(G,\mathbb{P}_G)\) and \(\mathcal{C}\!\it us\,(K,\mathbb{P}_K)\) induces a quasi-isometry between \(\mathcal{C}\!\it us\,(G,\mathbb{P}_G)\) and the convex hull of \(\Lambda_K\) which extends to a homeomorphism \(\Phi:\partial_{\mathbb{P}_G} G = \partial\mathcal{C}\!\it us\,(G,\mathbb{P}_G)\to\Lambda_K\). Since \(G\) is relatively hyperbolic, its action on \(\partial_{\mathbb{P}_G} G\) (and hence on \(\Lambda_K\)) is minimal and a geometrically finite convergence action. By Theorem 59, there is a finite index subgroup \(Q\) of \(G\) isomorphic to a finite index subgroup \(H\) of \(K\) and the isomorphism \(\varphi:Q\to H\) conjugates the actions on \(\Lambda_K\). Since the parings correspond to maximal parabolic subgroups, \(\varphi\) preserves the parings. \(\rule{.55em}{.55em}\)

6.3 Quasi-isometric rigidity of Kleinian groups↩︎

This section is devoted to the proof of Theorem 4. We apply the previous sections to build a Kleinian group from a quasi-isometry \(G\to K\). Theorems 45 and 61 and Lemma 19 produce a graph of groups structure for \(G\) with virtually Kleinian vertex groups. Then we use Proposition 55 to get a finite index subgroup which can be obtained by gluing fundamental groups of compact three-manifolds along their boundaries. This first leads us to prove the quasi-isometric rigidity of one-ended Kleinian groups:

Combining this theorem with Theorem 40 and Proposition 39, we obtain the quasi-isometric rigidity of Kleinian groups:

Proof. By Remark 17, \(G\) is quasi isometric to a torsion free minimally parabolic geometrically finite Kleinian group \(K\). Assume that \(K\) is not one-ended, in particular \(\Omega_K\neq\emptyset\). As explained in §4.2, when we decompose an irreducible three-manifold along a maximal union of essential discs, we get a graph of groups with trivial edge groups and one-ended or finite vertex groups. If we start with the Kleinian manifold \(M_K\), we get a graph of groups whose vertex groups are either trivial or one-ended Kleinian groups. Let \(G\) be a group quasi-isometric to \(K\). From Theorems 40 and 62 we get a graph of groups structure for \(G\) with finite edge groups such that every vertex group is either finite or has a finite index subgroup isomorphic to a one-ended geometrically finite Kleinian \(K_v\). Since \(\Omega_K\neq\emptyset\), \(\Omega_{K_v}\neq\emptyset\) hence \(\chi(\partial M_{K_v})<0\). From Proposition 39 we get that \(G\) has a finite index subgroup isomorphic to the fundamental group of an atoroidal three-manifold. Since \(G\) is quasi-isometric to \(K\) which is not one-ended, this three-manifold has non-empty boundary and hence is Haken. The conclusion follows from Theorem 14. \(\rule{.55em}{.55em}\)

Finally, Theorem 4 is a consequence of the latter together with Proposition 16. Now it remains to prove Theorem 62.

. Let \(G\) be a one-ended group quasi-isometric to a Kleinian group. By Theorems 45 and 61 and Lemma 19, \(G\) has a graph of groups structure \((\Gamma,\{G_v\},\{G_e\}, G_e\hookrightarrow G_{t(e)})\) with the following properties:

1. vertex groups are of essentially two types:

   1. pared groups with paring containing the adjacent edge groups and with a pared finite index subgroup isomorphic to a pared Kleinian group;

   2. virtually cyclic or virtually \(\mathbb{Z}^2\);

2. virtually Abelian vertex groups are not adjacent;

3. edge groups are virtually cyclic, and edge groups incident to an Abelian vertex group are commensurable (by Lemma 7).

By Theorem 54 type (a) vertices here are also type A for the definition of Proposition 55. Since Abelian groups are LERF (their subgroups are normal), type (b) vertices are also type B for the definition of Proposition 55. For each type (a) vertex \(v\) we pick a finite index normal pared torsion free Kleinian subgroup \(H_v<G_v\) and for each type (b) vertex \(w\) we pick a torsion free Abelian subgroup \(H_w<G_w\). By Proposition 55, \(G\) has a finite index subgroup \(G'\) which is the fundamental group of a finite graph of groups \({\mathcal{G}}'=(\Gamma',\{G_v'\},\{G_e'\}, G_e'\hookrightarrow G_{t(e)}')\) such that any vertex group is conjugate to a finite index subgroup of \(H_v\). In particular \({\mathcal{G}}'=(\Gamma',\{G_v'\},\{G_e'\}, G_e'\hookrightarrow G_{t(e)}')\) has the following properties:

1. Vertex groups are of essentially two types:

   1. pared torsion free geometrically finite Kleinian groups with paring containing the adjacent edges;

   2. Abelian groups isomorphic to \(\mathbb{Z}\) or \(\mathbb{Z}^2\).

2. Edge groups are cyclic, and edge groups incident to an Abelian vertex group all coincide.

Since any cyclic subgroup of \(\mathbb{Z}\) or \(\mathbb{Z}^2\) is contained in a unique maximal cyclic subgroup, we may assume that edge groups are primitive in type B vertex groups by Proposition 55.

This graph of groups structure for \(G'\) enables us to build a Kleinian group. To each vertex \(v\) we associate a compact three-manifold \(M_v\) with fundamental group \(G'_v\).

If \(G'_v\) is isomorphic to \(\mathbb{Z}\), \(M_v\) is a solid torus and for each adjacent edge \(e\), we pick an annulus \(A_e\) on the boundary of \(M_v\) such that the map \(i_*:\pi_1(A_e)\to\pi_1(M_v)\) induced by the inclusion is also the map \(G'_e\hookrightarrow G'_v\) defined by the graph of groups \({\mathcal{G}}'\). Since all those edge groups coincide and are primitive, we can choose the annuli corresponding to different edges to be embedded, disjoint and parallel.

If \(G'_v\) is isomorphic to \(\mathbb{Z}^2\), \(M_v\) is a thickened torus \(\mathbb{T}\times I\). For each adjacent edge \(e\), we pick an annulus \(A_e\subset\mathbb{T}\times\{0\}\) such that \(i_*:\pi_1(A_e)\to\pi_1(M_v)\) corresponds to \(G'_e\hookrightarrow G'_v\). Again we can choose the annuli corresponding to different edges to be embedded disjoint and parallel.

Otherwise, \(G'_v\) is isomorphic to a torsion free geometrically finite Kleinian group \(K_v\), \(M_v\) is the compact manifold whose interior is homeomorphic to \(\mathbb{H}^3/K_v\) and the incident edge groups and parabolic subgroups of \(G\) define a paring on \(\partial M_v\) corresponding to the parabolic subgroups of \(K_v\). In particular to each adjacent edge \(e\) is associated an annulus \(A_v\subset \partial M_v\).

Given an edge \(e=(v,v')\), we glue \(M_v\) and \(M_{v'}\) together along the annuli \(A_{\bar e}\subset M_v\) and \(A_e\subset M_{v'}\). The manifold thus produced does not depend on the map chosen to identify the annuli (up to homeomorphism). Doing this gluing for each edge, we get a compact three-manifold \(M\) whose fundamental group is \(G'\) [82].

By construction, \(M\) is irreducible. By Lemma 1, we just need to show that \(M\) is atoroidal to conclude with the hyperbolization theorem.

By property (6) of Theorem 45 any rank \(2\) virtually Abelian subgroup of \(G\) lies in a conjugate of a vertex group of \({\mathcal{G}}\), hence the same is true for \(G'\) and \({\mathcal{G}}'\). Let \(A\) be a rank \(2\) Abelian subgroup of \(G'\), up to conjugation, we have a vertex \(v\) such that \(A<G'_v\). If \(G'_v\) is isomorphic to \(\mathbb{Z}^2\) then \(M_v\) is a thickened torus \(\mathbb{T}\times I\) and \(A\) is a subgroup of \(\pi_1(\mathbb{T}\times\{1\})\). Since we have picked all the gluing annuli on \(\mathbb{T}\times\{0\}\), \(\mathbb{T}\times\{1\}\) is a boundary component of the final manifold \(M\). If \(G'_v\) is isomorphic to a Kleinian group, \(A\) is conjugate to a subgroup of the fundamental group of a torus \(T\subset\partial M_v\). By property (7) of Theorem 45 no gluing annulus can be homotoped in \(T\). It follows again that \(T\) is a boundary component of the final manifold \(M\). We have thus proved that \(M\) is atoroidal and the conclusion follows from the hyperbolization theorem. \(\rule{.55em}{.55em}\)

6.4 Quasi-isometric rigidity of three-manifold groups↩︎

We may now combine all our previous results together to deduce Theorem 2.

Proof.

Up to passing to a double cover, we may assume that \(M\) is orientable. We may also fill any sphere in \(\partial M\) with a ball. In particular \(M\) has non-positive Euler characteristic.

Notice that the only geometric three-manifolds with negative Euler characteristic are hyperbolic. If \(M\) is geometric, the conclusion follows from Theorems 11 and 63. Assume now that \(M\) is not geometric.

Let us first assume that \(G\) is one-ended. It follows that the fundamental group of \(M\) is also one-ended so that \(M\) is irreducible and \(\partial\)-irreducible, cf. § 2.2. In particular, by the Geometrization theorem, Theorem 10, \(M\) is Haken.

To make full use of the results of Kapovich and Leeb, let us consider the following coarser torus decomposition. The Euler characteristic decomposition \(T_{Eu}\subset T_B\) is the union of the components of \(T_B\) which either bound a component of \(M\setminus T_B\) with negative Euler characteristic or are parallel to a component of \(\partial M\). It follows from Theorem 41 that a component of \(M\setminus T_{Eu}\) either has zero Euler characteristic or is atoroidal (hence hyperbolic by Theorem 14 and Lemma 1). By Lemma 6 any quasi-isometry \(\widetilde{M}\to \widetilde{M}\) preserves the Euler characteristic decomposition so Theorem 42 still holds when we replace \(T_B\) with \(T_{Eu}\). It follows that \(G\) has a graph of groups structure \({\mathcal{G}}=(\Gamma,\{G_v\},\{G_e\}, G_e\hookrightarrow G_{t(e)})\) such that each vertex group \(G_v\) is quasi-isometric to the fundamental group \(Q_v\) of a compact three-manifold which is either hyperbolic or has zero Euler characteristic. Furthermore the quasi-isometry can be chosen to map the incident edge groups to conjugates of fundamental groups of boundary components. In particular, edge groups are virtually rank \(2\) Abelian by Theorem 7.

By Theorems 62, 12 and 3, for each vertex \(v\), \(G_v\) has a finite index subgroup \(H_v\) which is the fundamental group of a compact three-manifold \(N_v\) which is either hyperbolic or has zero Euler characteristic. Let \(\mathbb{P}_v\) be the collection of edge groups associated to the edges adjacent to \(v\), let \((H_v,\mathbb{Q}_v)\) be the adorned subgroup induced from \((G_v,\mathbb{P}_v)\) and assume that \(G_v\) is not Abelian. By Fact 33, \((H_v,\mathbb{Q}_v)\) is quasi-isometric to \((G_v,\mathbb{P}_v)\), which, by Theorem 42, is quasi-isometric to \((\pi_1(W(v)),P(v))\) where \(W(v)\) is a component of \(M\setminus T_{Eu}\) and \(P(v)\) is the collection of the fundamental groups of the tori of \(\partial \bar W(v)\). We equip \(N_v\) and \(W(v)\) with non-positively curved metrics, cf. Proposition 43. From the Švarc–Milnor lemma we get a quasi-isometry between the universal covers \(\widetilde{W}(v)\) of \(\bar W(v)\) and \(\widetilde{N}_v\) of \(N_v\). By [27], such a quasi-isometry maps boundary flats at bounded distance from boundary flats. It follows that each subgroup \(Q_i\) in \(\mathbb{Q}_v\) stabilizes, in \(\widetilde{N}_v\), a quasi-flat at bounded distance from a boundary flat \(F_i\) of \(\widetilde{N}_v\). An element of \(H_v\) acting as a covering transformation of \(\widetilde{N}_v\) maps boundary flats to boundary flats. Since the Hausdorff distance between two different boundary flats is infinite (see §4.3) and \(Q_i\) stabilizes a quasi-flat at bounded distance from \(F_i\), \(Q_i\) also stabilizes \(F_i\). Notice that \(Q_i\) might not be the whole stabilizer \(\mathrm{Stab}(F_i)< H_v\), but, since \(Q_i\) is quasi-isometric to an element of \(\mathbb{P}_v\), it is a rank \(2\) Abelian group and hence has finite index in \(\mathrm{Stab}(F_i)\) which is also a rank \(2\) Abelian group. On the other hand each boundary flat of \(\widetilde{W}(v)\) is stabilized by a subgroup conjugated to an element of \(P(v)\). We have thus established that the stabilizer of each boundary flat of \(\widetilde{N}_v\) has a finite index subgroup that is conjugated to an element of \(\mathbb{Q}_v\). By [105] up to replacing \(H_v\) with a finite index subgroup, and replacing \(N_v\) with the corresponding finite cover, we may assume that \(\mathbb{Q}_v\) is the collection of the fundamental groups of the tori of \(\partial N_v\).

By Theorem 54, vertices \(v\) with \(W(v)\) hyperbolic are type A for Proposition 55. Since edge groups are Abelian, it follows from [105] that vertices \(v\) such that \(W(v)\) has zero Euler characteristic are type B. It also follows from the definition of the Euler characteristic decomposition that if two vertices \(v\) and \(w\) are adjacent, then for at least one, say \(w\), \(W(w)\) is either hyperbolic or is a thickened torus that contains a component of \(\partial M\). In the latter case, \(G_w\) embeds into \(G_v\) and removing the vertex \(w\) (and its adjacent edge) would not change the group \(G\). The purpose of those redundant vertices is to guarantee that the collection \(P(v)\) of the fundamental groups of the tori of \(\partial \bar W(v)\) is also, up to conjugacy, the collection of adjacent edge groups in the graph of groups structure of \(\pi_1(M)\) associated to \(T_{Eu}\) (this convention simplifies previous statements). Thus we may assume that two type B vertices are not adjacent in the realm of Proposition 55.

Hence \({\mathcal{G}}\) satisfies the hypothesis of Proposition 55, so it follows that \(G\) has a finite index subgroup \(G'\) such that for any vertex \(v\), \(G'_v{=G'}\cap G_v\) is a normal finite index subgroup of \(H_v\). The subgroup \(G'_v\) is the fundamental group of a covering \(N'_v\) of \(N_v\). The subgroup \(G'\) inherits from \({\mathcal{G}}\) a graph of groups structure \({\mathcal{G}}'=(\Gamma',\{G'_v\},\{G'_e\}, G'_e\hookrightarrow G'_{t(e)})\) such that any vertex group \(G'_v\) is the fundamental group of a compact three-manifold \(N'_v\) and incident edge groups are conjugate to fundamental groups of boundary components.

For each edge \(e=(v,w)\) of \(\Gamma'\) we glue \(N'_v\) to \(N'_w\) along the components of their boundaries corresponding to \(e\). This produces a compact three-manifold whose fundamental group is \(G'\) and the conclusion follows.

When \(G\) is two-ended, then it is virtually cyclic and hence has a subgroup isomorphic to the fundamental group of a solid torus.

When \(G\) has infinitely many ends, the conclusion follows from the one-ended case together with Proposition 39 and Theorem 40. \(\rule{.55em}{.55em}\)

Let \(M\) be a compact orientable three-manifold. Let \(D\subset M\) be a maximal union of non isotopic essential spheres and discs. Given the compactification \(N\) of a component of \(M\setminus D\), we glue a ball along each component of \(\partial N\) that is a sphere. We call the resulting three-manifold a piece of \(M\setminus D\). Let \(T\) be the union of the balanced characteristic torus decompositions of the pieces of \(M\setminus D\) and let \(A\) be the union of the balanced annulus decompositions of the pieces of \(M\setminus (D\cup T)\). We call \(D\cup T\cup A\) the full decomposition of \(M\). Let \(G\) be a finitely generated group quasi-isometric to \(\pi_1(M)\). Assuming that no piece of \(M\setminus D\) is geometric, we can apply Theorems 40, 42 and 45, to get a graph of groups structure for \(G\) such that each vertex group is quasi-isometric to the fundamental group of a piece of \(M\setminus (D\cup T\cup A)\). Since no piece of \(M\setminus D\) is geometric, each piece of \(M\setminus (D\cup T\cup A)\) is either a pared acylindrical hyperbolic three-manifold, a Seifert manifold with boundary or an \(I\)-bundle over a compact surface with boundary. It follows then from Theorems 61, [115] and more classical results (see for example [116]) that each vertex group is commensurable to the fundamental group of a piece of \(M\setminus (D\cup T\cup A)\). Thus we have:

If a piece \(N\) of \(M\setminus D\) is geometric, we can still apply Theorem 40 and the characteristic torus (and annulus) decomposition of \(N\) is empty unless \(N\) is a Sol manifold. We deduce then from [115] and [42] that Theorem 65 is true when \(M\setminus D\) has some geometric pieces as long as those pieces are not Sol manifolds nor closed hyperbolic manifolds.

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