1 Introduction↩︎

Let \(G\) be a group and \(\mathbb{K}\) a field. A natural problem is to study relations between the group \(G\) and its group algebra \(\mathbb{K}[G]\). For instance, in 1953 Fox suggested that

“It seems reasonable to conjecture that a group ring \(\mathbb{Z}[G]\) can not have divisors of zero unless \(G\) has elements of finite order; this seems to be not an easy question.”([1], p.557).

This was also conjectured by Kaplansky in [2] and by Higman in his (unpublished) thesis (see [3]). It is equivalent to the statement that ideals generated by one element are free modules. In the early 60’s, Cohn went further (in [4]–[6]) and investigated rings in which ideals generated by any number of elements are free as \(\mathbb{K}[G]\)-modules (calling them free ideal rings, or firs for short), and showed that group algebras of free groups have this property. Soon after, Stallings proved his celebrated result on ends of groups [7] which implies no other group algebras do.

A related question is to describe, for a ring \(R\), the automorphism group \(\mathrm{GL}_n(R)\) of a free \(R\)-module. Denote by \(\text{GE}_n(R)\) the subgroup generated by elementary and diagonal matrices. Cohn also showed that for group algebras of free groups this is the entire automorphism group.

Bass [9] used these two theorems to show projective modules over the integral group ring of a free group, \(\mathbb{Z}[F]\), are free. This algebraic result has a striking topological consequence:

Our goal is to establish similar results for groups \(G\) acting on hyperbolic spaces. Earlier steps in this direction were taken in [11] and [12], where it was proved that ideals in \(\mathbb{K}[G]\) generated by one, respectively two elements are free if the minimum displacement of the action is large enough. Going further, we show in this paper that ideals generated by \(n\) elements are free if

(\(\mathcal{H}_{n}\)): \(G\) acts on a \(1\)-hyperbolic space with displacement more than \(100\log_2((n+1)!)\).

Under the same hypothesis, we also describe the automorphism groups of free, rank \(n\) modules.

Applications↩︎

To augment the description of \(\mathrm{GL}_n\), one needs to understand diagonal matrices, which amounts to describing the units in \(\mathbb{K}[G]\). This was already done in [11], where it was shown (assuming minimal displacement is greater than \(4\delta\)) that all units are trivial, i.e. have the form \(\lambda g\) for \(\lambda\in\mathbb{K}^*,g\in G\). In particular, for a finite field \(\mathbb{K}\) and finitely generated group \(G\) the unit group of \(\mathbb{K}[G]\) is finitely generated. Combining this with our theorem, we obtain an analogous result for \(\mathrm{GL}_n\).

In a different direction, a geometric consequence of our theorem is a lower bound for critical points of Morse functions of a given index on essential manifolds.

The theorem applies—for instance—if \(X=BG\) is a closed, riemannian manifold of curvature \(\leq -1\) and injectivity radius greater than \(50\log((n+1)!)\), or more generally if \(X\) has a map of non-zero degree to such a \(BG\).

Using Bass’s local-to-global method, we also obtain the following version of Corollary 1 for some \(2\)-dimensional hyperbolic groups (e.g. very high genus surface groups).

It is known ([13]) that one-relator groups have geometric dimenension at most two. In our setting of groups acting with large displacement on hyperbolic spaces, we can show that “few-relator” groups have cohomological dimension \(\leq 2\).

Finally, let us mention a consequence that can be thought of either as a generalization of Theorem 7 to higher dimensions or as a generalization of (the \(X=BG\) case of¹) Theorem 5 to hyperbolic groups.

Algorithms↩︎

The proof of the first part of Theorem 3 is based on an algorithm which can be seen as a geometric version of the euclidean algorithm. To describe it, let us first sketch the approach to the “only if” direction of Theorem 1 given in Cohn’s book [6]. It consists of two distinct steps:

• Fix a basis for the free group \(F\), let \(F_+\) be the (monoid of) non-negative words in this basis and denote by \(\mathbb{K}[F_+]\subset\mathbb{K}[F]\) the subring of the group algebra generated by these words. Cohn first shows that ideals in \(\mathbb{K}[F_+]\) are free. (Corollary 2.5.2 and Theorem 2.4.6)

• He then gives a localization procedure for passing from \(\mathbb{K}[F_+]\) to \(\mathbb{K}[F]\) that preserves the free ideal property. (Corollary 7.11.8)

Cohn measures size via filtrations. Recall that a filtration on a ring \(R\) is a map \(|\cdot| : R\rightarrow \mathbb{N}\cup\{-\infty\}\) such that \(|0|=-\infty\), \(|1| = 0\), and for all \(x,y\in R\) \(|x-y|\leqslant\max(|x|,|y|),\) \(|xy|\leqslant|x| + |y|.\)

Given a generating set for a group \(G\), the group algebra \(\mathbb{K}[G]\) has a natural filtration by word length \(l(g)\) of elements \(g\) in \(G\): \[\left|\sum_{g\in G}\lambda^g\cdot g\right|:=\max_{\lambda^g\not=0}l(g).\] On the ring \(\mathbb{K}[F_+]\) of positive words in the free group, this filtration satisfies \(|xy|=|x|+|y|\).

The relevant notion of dependence says that some linear combination has smaller size than the maximum dictated by its terms: A family \(\xi_1, \dots, \xi_n\) in \(R\) is \(|\cdot|\)-dependent if there is a non-zero \((\alpha_1, \ldots, \alpha_n)\in R^n\) such that \[\left|\sum_i \alpha_i \xi_i\right|<\max_i\left|\alpha_i\xi_i\right|.\]

The first step is accomplished via the following algorithmic theorem.

For a finitely generated ideal in \(\mathbb{K}[F_+]\), one can repeatedly apply this theorem to decrease the size of members of a finite generating set until one arrives at a generating set that does not satisfy any dependence relations and hence forms a basis, showing the ideal is free. Doing a bit more work, Cohn uses this theorem to show infinitely generated ideals in \(\mathbb{K}[F_+]\) are free, as well (Thm. 2.4.6). Finally, Cohn’s localization proceedure shows the same is true for \(\mathbb{K}[F]\).

In [14], Hog-Angeloni gave a beautiful, geometric version of this algorithm that applies to the entire group algera \(\mathbb{K}[F]\). It can be used to bypass the localization step if one is interested exclusively in finitely generated ideals. Hog-Angeloni’s proof goes by looking at the action of \(F\) on its Cayley graph, which is a tree \(\mathcal{T}\). It uses the same notion of dependence as Cohn’s proof but the natural notion of size of group ring elements relevant to her argument is the diameter \[\mathrm{diam}\left(\sum_{g\in G}\lambda^g\cdot g\right):=\max_{\lambda^g\not=0\not=\lambda^h}l(g^{-1}h).\]

An attentive reader of [14] can check that Hog-Angeloni does not use that the group is free nor that the tree is a Cayley graph, but only the fact that the group acts freely on an \(\mathbb{R}\)-tree. We build on her geometric approach, replacing \(F\) acting on the tree \(\mathcal{T}\) by a group \(G\) that acts on a hyperbolic space \(\mathcal{H}\) with large minimum displacement. In our approach we will use the natural filtration on \(\mathbb{K}[G]\) obtained from the action of \(G\) on the hyperbolic space \(\mathcal{H}\).

Now, we can state our hyperbolic version of Hog-Angeloni’s theorem.

Acknowledgements↩︎

We would like to thank Misha Gromov for his interest in this work and for mentioning the possibility of extending our results to essential manifolds. G.A. would like to thank the Max Planck Institut für Mathematik for its hospitality and financial support. We would like to thank the referee for pointing out a gap in an earlier version of the proof of Theorem 40, for their useful comments, and for highlighting an interesting perspective concerning the structure of the ring \(\mathbb{K}[\mathbb{Z}\times G]\) (see 5.3).

2 Hyperbolic preliminaries↩︎

Fix a \(\delta\)-hyperbolic metric space \(\mathcal{H}\) with basepoint \(o\). See [15],[16], or [17] for background on hyperbolicity.

Radii and centers↩︎

Let \(\mathcal{X}\) be a bounded subset in the metric space \(\mathcal{H}\). Denote by \[r(\mathcal{X}):=\inf\{r\midthere iscfor whichB(c,r)\supset\mathcal{X}\}\] the infimum of radii of closed balls containing \(\mathcal{X}\). We call it the radius of \(\mathcal{X}\). If \(\mathcal{H}\) is a proper metric space (i.e. if its closed, bounded sets are compact), then this infimum is realized and there is a closed ball of radius \(r(\mathcal{X})\) containing \(\mathcal{X}\). If \(\mathcal{H}\) is a complete, \(\mathrm{CAT}(0)\) space, then there is a unique such ball. In general, we only have for any positive \(\epsilon\) a closed ball \(B(c,r(\mathcal{X})+\epsilon)\) containing \(\mathcal{X}\). We call such a \(c\) an \(\epsilon\)-center of \(\mathcal{X}\).

If \(\mathcal{H}\) is an arbitrary metric space, one can embed \(\mathcal{H}\) isometrically in a space in which every finite subset has a \(0\)-center. Let \(\omega\) be a non-principal ultrafilter; the ultralimit \(\mathcal{H}^\omega\) of the constant sequence \((\mathcal{H}, x_0, d)\) satisfies this property, as can be easily checked. Furthermore, if \(\mathcal{H}\) is geodesic and \(\delta\)-hyperbolic then the ultralimit \(\mathcal{H}^{\omega}\) is, as well, and every isometry of \(\mathcal{H}\) extends to an isometry of \(H^\omega\) with the same infimum displacement.

Gromov products↩︎

Recall ([15]–[17]) the definition of the Gromov product \[\left<p,q\right>_r:={1\over 2}(|p-r|+|q-r|-|p-q|).\] To simplify notation, recall that the distance from \(p\) to the origin \(o\) is denoted by \(|p|\) and set \(\left<p,q\right>:=\left<p,q\right>_o\). First, we give an estimate for the Gromov product of an \(\epsilon\)-center of \(\mathcal{X}\) with a point of \(\mathcal{X}\) that follows directly from the triangle inequality.

Thin triangles and projections in hyperbolic spaces↩︎

In a \(\delta\)-hyperbolic space, any geodesic triangle is \(\delta\)-thin. This means that if \([r,p,q]\) is a triangle then two (oriented) geodesics \([r,p]\) and \([r,q]\) parametrized by arc length \(p(t),q(t)\) remain \(\delta\)-close (i.e. \(|p(t)-q(t)|\leq\delta\)) until \(t=\left<p,q\right>_r\). We will repeatedly use the following consequence of this: Given a segment \([a,b]\) and a point \(x\in\mathcal{H}\), the projection of \(x\) to \([a,b]\) is the point \(x'\) on \([a,b]\) such that \(|a-x'|=\left<b,x\right>_a\).² Then, for points \(y_1\) in the initial subsegment \([a,x']\) of \([a,b]\) we have a \(\delta\)-converse to the triangle inequality (see Figure 1): \[\label{conversetriangle} |x-y_1|+|y_1-a|\leq|x-a|+\delta,\tag{1}\] while for points \(y_2\) in the subsegment \([x',b]\) we have \[|x-y_2|+|y_2-b|\leq|x-b|+\delta.\]

Diameter vs radius↩︎

First, we note that in a \(\delta\)-hyperbolic space—just like in a tree—the diameter and radius are closely related. More precisely,

Center vs midpoint↩︎

Next, we observe that an \(\epsilon\)-center is \((2\delta+\epsilon)\)-close to the midpoint of any diameter realizing segment.

In particular, any two \(\epsilon\)-centers of a finite set are \((4\delta+2\epsilon)\)-close to each other and—by Lemma 2—the midpoints of any two diameters are \(3\delta\)-close to each other.

An equivariant choice of centers↩︎

Therefore, as long as the displacement is greater than \(3\delta\) we can choose for each finite subset \(\mathcal{X}\subset \mathcal{H}\) an \(\epsilon\)-center \(c(\mathcal{X})\) such that \(c(g\mathcal{X})=gc(\mathcal{X})\) for each \(g\in G\).

Distance from origin to center↩︎

Now, we estimate the distance from the origin to an \(\epsilon\)-center of a set \(\mathcal{X}\) in terms of its radius and \(|\mathcal{X}|\).

Diameter of intersection of two balls↩︎

Another property of \(\delta\)-hyperbolic spaces we will need is a bound on the diameter of the intersection of two balls. The diameter of a set \(\mathcal{X}\) (not necessarily finite) is \(\inf_{a,b\in\mathcal{X}}|a-b|\).

Gromov product inequality↩︎

Finally, we recall a key inequality that will be used to estimate distances:

3 Extremal graphs↩︎

Let \(V\) be a finite set and suppose we have a family³ \(Y=(\mathcal{Y}_v)_{v\in V}\) of bounded subsets of \(\mathcal{H}\). For subsection 3.1, we fix \(\epsilon\) arbitrarily small (say \(\epsilon<<\delta\)) and chose for every set \(\mathcal{Y}_v\) an \(\epsilon\)-center \(c_v\). (If \(\mathcal{H}\) is proper or complete CAT(0), we set \(\epsilon=0\) and let \(c_v\) be a center of \(\mathcal{Y}_v\).) Set \(d=|\cup_{v\in V}\mathcal{Y}_v|=\max_{v\in V}|\mathcal{Y}_v|\). Let \(\mu\) be a fixed parameter. A point \(p\in\cup_{v\in V}\mathcal{Y}_v\) satisfying \(|p|\geq |\cup_{v\in V}\mathcal{Y}_v|-\mu\) is called a \(\mu\)-extremal point (of \(Y\)). A \(0\)-extremal point will also be called an extremal point.

3.1 The graph \(\Gamma_{\mu}(Y)\)↩︎

Let us first define the \(\mu\)-extremal graph \(\Gamma_{\mu}(Y)\) of the family \(Y\). The vertices of \(\Gamma_{\mu}(Y)\) are the indices \(v\in V\) such that \(\mathcal{Y}_v\) contains a \(\mu\)-extremal point (of \(Y\)), and there is an edge between \(v\) and \(w\) for each \(\mu\)-extremal point in \(\mathcal{Y}_v\cap\mathcal{Y}_w\).

Distance between centers↩︎

For each vertex \(v\in V\), let \(r_v\) be radius of the set \(\mathcal{Y}_v\). We estimate the distance between two centers \(c_v\) and \(c_w\) in terms of the distance between the vertices \(v\) and \(w\) in the graph \(\Gamma_{\mu}(Y)\). By a (simplicial) path between \(v\) and \(w\), we mean a sequence \(v=v_0,v_1,\dots,v_m=w\) of vertices, such that for each \(i\), \(v_i\) and \(v_{i+1}\) are connected by an edge \(e_i\) in the graph.

Colors↩︎

Consider a partition \(V=V_1\sqcup\dots\sqcup V_n\) into \(n\) subsets called colors such that any two vertices of the same color have the same radius (if \(v,w\in V_i\) then \(r_v=r_w\)). Denote by \(r_i\) the radius of a set of color \(i\).

Bounding the diameter of \(\Gamma_{\mu}(Y)\)↩︎

Next, we will give an upper bound on the diameter of a component of an extremal graph in terms of the number \(n\) of colors. The result will be proved by induction on the number of colors \(n\). In fact, we prove something stronger, namely an upper bound on the length of an embedded path in the graph.

Recall that \(r_1\geq\dots\geq r_n\) so that \(r_{\max(V)}=r_1\).

3.2 The graph \(\Gamma_{\mu}(\Xi)\)↩︎

We will apply Proposition 12 and Corollary 3 in the following situation. Let \(\Xi=(\xi_v)_{v\in V}\) in \(\mathbb{K}[G]\) be a finite family of group ring elements, such that no two are \(\mathbb{K}\)-scalar multiples of each other. Recall that the (geometric) support of \(\xi_v=\sum_{g\in G}\xi_v^g\cdot g\) is the set \(\mathcal{X}_v=\{g\cdot o\mid \xi_v^g\not=0\}\). Let \(X=(\mathcal{X}_v)_{v\in V}\) be the family of these supports, and set \(\Gamma_{\mu}(\Xi):=\Gamma_{\mu}(X)\). In this situation, there is a canonical partition of the index set \(V\) into colors \(V=V_1\sqcup\dots\sqcup V_n\) defined as follows. We declare that \(v\) and \(w\) have the same color if and only if there is \(\lambda\in\mathbb{K}^*\) and \(g\in G\) such that \(\xi_v=\lambda g\xi_w\). Note that if \(v\) and \(w\) have the same color then \(g\mathcal{X}_v=\mathcal{X}_w\) and hence \(r_v=r_w\). Let \(r_i\) be the radius of elements with color \(i\).

Components of \(\Gamma_{\mu}(\Xi)\) and weak relations↩︎

A finite collection of group ring elements \((\xi_v)_{v\in V}\) defines a \(\mu\)-relation if \[\label{murelation} \left|\sum_{v\in V}\xi_v\right|<\max_{v\in V}|\xi_v|-\mu.\tag{2}\] The components of \(\Gamma_{\mu}(\Xi)\) help us keep track of such \(\mu\)-relations.

4 Proof of Theorem 11↩︎

We inductively define \(\delta_n\) by \(\delta_0=0\), \(\delta_k=\delta_{k-1}+(2(4+\log_2(k+1))+7)\delta\). Then, \[\begin{align} \delta_n&=&(15n+2\log_2((n+1)!))\delta. \end{align}\]

4.1 Restatement of the main theorem↩︎

We will now restate Theorem 11 in terms of colors and \(\mu\)-relations. We assume that \(\xi_1,\dots,\xi_n\) are elements in the group algebra \(\mathbb{K} [G]\) so that there exist \(\alpha_1, \ldots, \alpha_n\) in \(\mathbb{K} [G]\), not all zero, satisfying \(| \sum \alpha_i \xi_i | < \max_i | \alpha_i \xi_i | - \delta_n\). Recall that two elements \(\xi_1\) and \(\xi_2\) of \(\mathbb{K} [G]\) have the same color if there exists a trivial unit \(\lambda g\) in \(\mathbb{K} [G]\) such that \(\xi_1 - \lambda g \xi_2 = 0\). Certainly if two of the \(\xi_i\) have the same color, the conclusion of the theorem follows. So, from now on, we assume that the \(\xi_i\) have different colors. Then, the index set \(\{ 1, \ldots, n \}\) can be used as the set of colors. For each \(\alpha_i = \sum_{g \in G} \alpha^g_i \cdot g\), we let \(A_i = \{g \in G \mid \alpha^g_i \neq 0\}\) be its (algebraic) support in the group \(G\). The group ring elements \((\alpha^g_i g \xi_i)_{g \in A_i, 1 \leq i \leq n}\) are distinct, no two of them are scalar multiples of each other, and for \(i \neq j\) they do not have the same color. We rename them \((\xi_v)_{v \in V}\) which is a collection of elements in \(\mathbb{K} [G]\) having \(n\) colors. With this notation, as the \(\xi_v\) are exactly the translates \(\alpha_i^g\xi_i\), the sum \(\sum_i\left(\sum_{g\in A_i}\alpha_i^g\right)\xi_i\) is now written as \(\sum_{v\in V}\xi_v\) and the hypothesis \(| \sum \alpha_i \xi_i | < \max_i | \alpha_i \xi_i | - \delta_n\) implies \[\left| \sum_{v \in V} \xi_v \right| < \max_{v \in V} \left| \xi_v \right| - \delta_n.\] Recall that in this situation we say that the family \((\xi_v)_{v \in V}\) defines a \(\delta_n\)-relation. Under this hypothesis, we will show that there exists a vertex \(v_*\) in \(V\) and a subset \(S \subset V\) made of elements of different colors from the color of \(v_*\) such that \(\mathrm{diam} (\xi_{v_*} + \sum_{v \in S} \xi_v) < \mathrm{diam} (\xi_{v_*}) - \delta\).

Summing up, Theorem 11 will follow from

4.2 The case of trees↩︎

We keep the notations of the previous paragraph unchanged, but assume that the space \(\mathcal{H}\) is a tree. This is the case \((\delta = 0)\) treated by Hog-Angeloni in [14]. We slightly rephrase her proof.

Recall that \(\Gamma_0 = \Gamma_0 (\Xi)\) is the graph whose vertices are the elements \(v\) in \(V\) such that if \(\mathcal{X}_v\) is the support of \(\xi_v\), then \(\mathcal{X}_v\) contains an extremal point, i.e. \(| \xi_v | = d = \max_{v \in V} | \xi_v |\), and there is an edge between \(v\) and \(w\) for each extremal point in \(\mathcal{X}_v\cap\mathcal{X}_w\). The colors \(\{ 1, \ldots, n \}\) are organized so that the \(r_i\) (the radii of the \(\mathcal{X}_i\)) are in decreasing order: \(r_1 \geq \cdots \geq r_n\).

Let \(\hat{\Gamma}_0\) be a component of \(\Gamma_0\) containing a vertex \(v_*\) with color \(1\), the color of largest radius. By Lemma 8, this graph defines a \(0\)-relation \[\left| \sum_{v \in \hat{\Gamma}_0} \xi_v \right| < \max_{v \in \hat{\Gamma}_0} | \xi_v | .\] By Corollary 4, \(v_*\) is the unique vertex in \(\hat{\Gamma}_0\) of color \(1\). So, removing \(v_*\) from \(\hat{\Gamma}_0\) and setting \(S = \hat{\Gamma}_0 - \{v_*\}\), we see that to conclude the proof of the theorem we need to estimate the diameter of the group ring element \(\sum_{v\in\hat{\Gamma}_0}\xi_v=\xi_{v_*}+\sum_{v\in S}\xi_v\).

Note that for any two adjacent vertices \(v\) and \(w\) in \(\hat{\Gamma}_0\), both centers \(c_v\) and \(c_w\) lie on a geodesic \([o, p]\) from the origin to an extremal point. The oriented geodesics \([o, c_v]\) and \([o, c_w]\) coincide along a segment of length \(\min \{|c_v |, |c_w |\} \geq d - r_1 = | c_{v_*} | .\) By following a path from \(v_*\) to \(v\) in the graph \(\hat{\Gamma}_0\) and noting that \(r_1 \geq r_w\) for every vertex \(w\) along that path, we conclude that \(c_{v_*}\) belongs to every geodesic \([o, c_v]\), and \[|c_{v_*} - c_v | = r_1 - r_v .\] For every \(v,\) the support of \(\xi_v\) is contained in the closed ball \(B (c_{v_*}, r_1)\), so the same is true for the support of \(\sum_{v \in\hat{\Gamma}_0} \xi_v\). Moreover, because of the \(0\)-relation, this support has no \(0\)-extremal points, i.e. it is contained in the ball \(B (o, d-\alpha)\) for some \(\alpha > 0\). So, applying Lemma 5, we see that the the diameter of the support of \(\sum_{v\in\hat{\Gamma}_0}\xi_v\) is at most \[d - \alpha + r_1 - |o - c_{v_*} | = 2 r_1 -\alpha= \mathrm{diam} (\xi_{v_*}) - \alpha .\] This finishes the proof.

4.3 Proof of Theorem 13 for hyperbolic spaces↩︎

In order to prove Theorem 13, we keep the notations of 4.1 unchanged and proceed by induction on the number \(n\) of colors. Let us fix some notation. We are given a collection of elements \(\Xi=(\xi_v)_{v\in V}\) defining a \(\delta_n\)-relation. Denote \(d=\max_{v\in V}|\xi_v|\). For any \(\mu\geq 0\) we abbreviate \(\Gamma_{\mu}=\Gamma_{\mu}(\Xi)\), so that vertices of \(\Gamma_{\mu}\) are those \(v\in V\) for which \(|\xi_v|\geq d-\mu\).

Base case↩︎

If there is one color, then \(\delta_1=17\delta\) and the minimum displacement is greater than \(4\delta_1+12\delta\). So, Corollary 4 applies and implies that connected components of \(\Gamma_{\delta_1}\) are points, and there are no \(\delta_1\)-relations.

Inductive step↩︎

Suppose now that we know the theorem for \(n-1\) colors and want to prove it for \(n\). By the inductive hypothesis, we may assume that \(\Xi\) satisfies the following “minimality” condition:

• No subset \(W\subset V\) consisting of fewer than \(n\) colors defines a \(\delta_{n-1}\)-relation.

Since we will consider different values of the parameter \(\mu\), it is useful to keep in mind that for \(\mu<\mu'\) the graph \(\Gamma_{\mu}\) is a subgraph of \(\Gamma_{\mu'}\). In particular, for a vertex \(v_0\in\Gamma_0\) (corresponding to an element \(\xi_{v_0}\) whose support contains an extremal point) we have inclusions \[v_0\in\Gamma_0\subset\Gamma_{\delta_{n-1}}\subset\Gamma_{\delta_n}.\] Denote by \(\hat{\Gamma}_{\delta_{n-1}}\) and \(\hat{\Gamma}_{\delta_n}\) the connected components of \(\Gamma_{\delta_{n-1}}\) and \(\Gamma_{\delta_n}\) containing the vertex \(v_0\). Since \(\Xi\) defines a \(\delta_n\)-relation (and \(\delta_n>\delta_{n-1}\)) Lemma 8 implies that \(\hat{\Gamma}_{\delta_{n-1}}\) defines a \(\delta_{n-1}\)-relation, so minimality implies \(\hat{\Gamma}_{\delta_{n-1}}\) contains all \(n\) colors.

Large and small colors↩︎

Order the colors in decreasing order, \(r_1\geq\dots\geq r_n\), and let \(k\) be the largest integer for which \(r_k\geq r_1-\delta_n\). We call \(\{1,\dots,k\}\) the large colors and the rest small colors. The minimal displacement \(> 4\delta_n+(10+2n)\delta\) assumption implies, by Corollary 4.2, that each large color appears exactly once in the connected graph \(\hat{\Gamma}_{\delta_n}\). Thus, we can identify the set of large colors \(\{1,\dots,k\}\) with a set of \(k\) vertices \(\{v_1,\dots,v_k\}\) in \(\hat{\Gamma}_{\delta_n}\). Since each color appears in \(\hat{\Gamma}_{\delta_{n-1}}\), we conclude \[\{v_1,\dots,v_k\}\subset\hat{\Gamma}_{\delta_{n-1}}\subset\hat{\Gamma}_{\delta_n}.\]

Centers↩︎

Next, we compare the positions of the centers of the sets \(\mathcal{X}_v\). We will prove

We need the following lemmas.

The correction term \((4+\log_2(n+1))\delta\) in Proposition 14 recurs throughout the rest of the proof so, to save space and improve readability, set \(L(n)=4+\log_2(n+1)\).

Note that for any \(w\in\hat{\Gamma}_{\delta_n}\) we have⁴ \[\label{centersize} |c_w|\geq d-\delta_{n-1}-r_1.\tag{3}\] So, since \(\delta_n\geq\delta_{n-1}\), we can pick \(c'_w\in [o,c_w]\) such that \(|c'_w|=d-{\delta_n+\delta_{n-1}\over 2}-r_1-L(n)\delta\).

Bounding the diameter of support↩︎

Recall that the family \(\Xi\) defines a \(\delta_n\)-relation, so the support of \(\sum_{v\in\hat{\Gamma}_{\delta_n}}\xi_v\) is contained in the ball \(B(o,d-\delta_n)\). Let \(c^*\) be the point in \([o,c_{v_1}]\) such that \(|c^*|=d-\delta_{n-1}-r_1-L(n)\delta\). Then—by definition—we have \(|c^*-c'_{v_1}|={\delta_n-\delta_{n-1}\over 2}\). Next, we will show that the ball \(B(c^*,r_1+(L(n)+2)\delta)\) also contains the support of \(\sum_{v\in\hat{\Gamma}_{\delta_n}}\xi_v\):

We’ve shown that the support of \(\sum_{v\in\hat{\Gamma}_{\delta_n}}\xi_v\) is contained in the intersection of balls \(B(o,d-\delta_n)\cap B(c^*,r_1+(L(n)+2)\delta)\). Therefore, Lemma 5 implies the diameter of the support is \[\begin{align} &\leq&(d-\delta_n)+(r_1+(L(n)+2)\delta)-(d-r_1-\delta_{n-1}-L(n)\delta)+2\delta\\ &=&(2r_1-2\delta)-\delta_n+\delta_{n-1}+(2L(n)+6)\delta\\ &\leq&\mathrm{diam}(\xi_{v_1})-\delta, \end{align}\] where for the last inequality we’ve used the fact that \(2r_1\leq\mathrm{diam}(\xi_{v_1})\) by Lemma 2 and the inductive description of \(\delta_n\). Since \(v_1\) is the unique vertex of color \(1\) in \(\hat{\Gamma}_{\delta_n}\), this establishes the theorem.

5 Applications↩︎

We now apply the algorithm. The main hypothesis in this section is:

(\(\mathcal{H}_{n}\)): \(G\) acts on a \(1\)-hyperbolic space with displacement more than \(100\log_2((n+1)!)\).

Let \(\mathrm{E}_n (\mathbb{K} [G])\) be the subgroup of elementary matrices in \(\mathrm{GL}_n (\mathbb{K} [G])\). We begin with a linear algebraic lemma.

5.1 Freeness↩︎

Lemma 14 enables the study of finitely generated submodules of free modules.

From this theorem and Stallings result on groups with infinitely many ends, we can also deduce:

5.2 The group \(\text{GL}_n (\mathbb{K} [G])\)↩︎

Recall that \(\text{GE}_n (\mathbb{K} [G])\) is the subgroup of \(\text{GL}_n (\mathbb{K} [G])\) generated by elementary and diagonal matrices.

Theorems 15.1 and 16 together establish Theorem 3 from the introduction.

5.3 Submodules of free \(\mathbb{Z} [G]\)-modules↩︎

Until now we’ve been studying the group algebra \(\mathbb{K} [G]\) with coefficients in a field \(\mathbb{K}\). Our next goal is to extend freeness results to the integral group ring \(\mathbb{Z} [G]\). We follow the method Bass used in [9] to show projective \(\mathbb{Z}[F]\)-modules are free, but with two differences. First, the article [9] studies group rings with coefficients in a principal ideal domain. We restrict ourselves to the ring \(\mathbb{Z}\). Second, the main hypothesis of [9] is that \(M\) is a projective module. Instead, we use

• The module \(M\) embeds as a submodule of a free module \(\oplus\mathbb{Z} [G]\) such that the quotient \((\oplus \mathbb{Z}[G])/M\) is torsion free as an abelian group.

This condition is probably known to specialists but hard to locate in the literature. It is a weakening of the more familiar “projective”. Indeed, if \(M\) is a projective module, there exists a module \(N\) such that \(M \oplus N = \oplus \mathbb{Z} [G]\) is a free module, in particular the quotient \(N\) is torsion free. It is useful for obtaining topological consequences thanks to the following important example.

Here is the main theorem of this subsection.

To ellucidate the role of \((\star)\) in the proof, we recall some basics on abelian groups.

On abelian groups and torsion↩︎

For an abelian group \(A\), let \(A_{\mathbb{Q}}:=A\otimes_{\mathbb{Z}}\mathbb{Q}\) be its rationalization and \(A_p:=A\otimes_{\mathbb{Z}}\mathbb{F}_p\) its mod \(p\) reduction. An inclusion of abelian groups induces homomorphisms of rationalizations and mod \(p\) reductions. The latter may no longer be an inclusion. In our proof, condition (\(\star\)) will be relevant because it implies injectivity of the induced map on mod \(p\) reductions via the last part of the following lemma which summarizes basic properties of rationalizations and mod \(p\) reductions that we will need.

A ‘local-to-global’ principle for rings↩︎

We can now prove Theorem 18. Note that \(\mathbb{Z}[G]\) is torsion-free as an abelian group, so any submodule of \(\oplus\mathbb{Z}[G]\) is, as well. Moreover, the rank of a finitely generated free \(\mathbb{K}[G]\)-module is the dimension of the vector space of coinvariants \(\mathbb{K}[G]^m\otimes_{\mathbb{K}[G]}\mathbb{K}=\mathbb{K}^m\). So, Theorem 18 is a consequence of Theorem 15.2, Theorem 16 and the \(R=\mathbb{Z}[G]\) case of the following purely ring theoretic proposition.

5.4 Chain complexes, cell decompositions, and Morse theory↩︎

Recall that the cohomological dimension of the group \(G\), denoted \(\mathrm{cd}(G)\), is the minimal length of a free \(\mathbb{Z}[G]\)-resolution of \(\mathbb{Z}\) (see [25]).

Essential maps↩︎

If \(\mathbb{K}\) is a field, the same result is true with \(\mathbb{Z}\) replaced by \(\mathbb{K}\) and cohomological dimension by \(\mathbb{K}\)-cohomological dimension (\(=\) minimal length of a free \(\mathbb{K}[G]\)-resolution of \(\mathbb{K}\)), and easier to prove as we don’t need Theorem 18, but only Theorem 15.2. In fact, in the setting of \(\mathbb{K}\)-cohomological dimension we have the following more general result suggested by a question of Gromov.

Let \(G\) be a classifying space for the group \(G\). For a cell complex \(X\), we say that a cellular map \(X\rightarrow BG\) is \(d\)-essential (with local coefficients) if there is a \(\mathbb{K}[G]\)-module \(V\) such that the induced map \(H^{d}(BG;V)\rightarrow H^{d}(X;V)\) is non-zero. For the sake of brevity, we will omit “with local coefficients” from now on. We will call a \(d\)-manifold \(X\) essential if the map \(X\rightarrow B\pi_1(X)\) is \(d\)-essential.

Let us spell out a consequence of a special case of Theorem 21, where we suppose that \(X^d\) is a closed \(d\)-manifold. Recall that a Morse function gives a cellular decomposition of a manifold with one cell of dimension \(k\) for each point of index \(k\) ([26], or [27] for a detailed proof). Therefore we deduce

5.5 Dimensions, few-relator groups, and \(2\)-complexes↩︎

Concerning few-relator groups, we get the following.

Concerning the topology of presentation complexes, we get the following.

References↩︎

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1. See also Theorem 21.↩︎

2. This is also the point such that \(|b-x'|=\left<a,x\right>_b\), since \(\left<a,x\right>_b+\left<b,x\right>_a=|a-b|\).↩︎

3. Repetitions in \(Y=(\mathcal{Y}_v)_{v\in V}\) are allowed, i.e. we may have \(\mathcal{Y}_v=\mathcal{Y}_w\) even if \(v\not=w\).↩︎

4. Either \(w\) is small, in which case \(|c_w|\geq d-\delta_{n}-r_w\geq d-r_1\) or \(w\in\hat{\Gamma}_{\delta_{n-1}}\) and then \(|c_w|\geq d-\delta_{n-1}-r_w\geq d-\delta_{n-1}-r_1\).↩︎

5. The reason is that if a chain complex \(C\) of finitely generated free \(\mathbb{F}_2[G]\)-modules is not acyclic, the its \(\mathbb{F}_2[G]\)-hom dual \(C^*\) is not acyclic either, since \(C^{**}=C\).↩︎

6. A finitely generated \(R\)-module is free of unique rank if it is isomorphic to \(R^m\) for a unique \(m\).↩︎

7. For an element \(a\) in a torsion-free abelian group, if there is an element \(a_0\) such that \(a=pa_0\), then there is a unique such element. We call this element \({a\over p}\).↩︎