October 02, 2025
We find surface subgroups in certain one-relator groups with torsion and use this to deduce a profinite criterion for a word in the free group to be primitive.
A longstanding question often attributed to Gromov asks whether every one-ended hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface, which from now on we will refer to as containing a surface subgroup. This has generated a lot of work on finding surface subgroups in various classes of hyperbolic groups. One of the most famous results in this vein is the existence of surface subgroups in fundamental groups of closed hyperbolic 3-manifolds [1]. Recall that a hyperbolic group is said to be rigid if it does not admit a nontrivial splitting with a virtually cyclic edge group. Another milestone in the theory is [2], where Wilton proved that a one-ended hyperbolic group without \(2\)-torsion \(\Gamma\) contains either a quasi-convex surface subgroup or a quasi-convex rigid subgroup, hence reducing the question of existence of surface subgroups for hyperbolic groups without 2-torsion to the question for one-ended rigid hyperbolic groups. We refer the reader to [2] and the references therein for more background.
The aim of this note is to prove the following:
Theorem 1. Let \(F_k\) be the non-abelian free group with free basis \(\{x_1, \dots x_k\}\) and let \(w\) be a word in the \(x_i\) which isn’t primitive. Then there is some positive integer \(d\) such that, for sufficiently large \(n\) which are multiples of \(d\), the one-relator group \(G_n:=F_k/\langle\langle w^n\rangle\rangle\) contains a quasi-convex surface subgroup.
Recall that a word \(w\) in \(F_k\) is said to be primitive if \(F_k\) splits as a free product \(\langle w \rangle *F_{k-1}\). The assumption that \(w\) isn’t primitive can’t be removed since when \(w\) is primitive the corresponding one-relator groups are virtually free.
There has also been substantial interest in characterising when a word is primitive. Whitehead famously gave an algorithm that can be used to decide whether a given word is primitive [3]. Later work of Puder and Parzanchevski showed that an element \(w\) of \(F_k\) which is primitive in \(\widehat{F_k}\) is already primitive in \(F_k\) [4], which was later reproved and generalised by Wilton [2]. Using 1 we can give another profinite characterisation of when a word is primitive:
Corollary 2. Let \(F_k\) be the non-abelian free group with free basis \(\{x_1, \dots x_k\}\) and let \(w\) be a word in the \(x_i\). The following are equivalent:
\(w\) is primitive;
For all sufficiently large \(n\) the one-relator group \(G_n:=F_k/\langle \langle w^n \rangle \rangle\) has the same profinite completion as the group \(K_n:=F_{k-1}*\mathbb{Z}/n\).
Recall that Remeslennikov’s famous question [5] asks whether all finitely generated residually finite groups that have the same profinite completion as a free group are in fact free. A positive answer to Remeslennikov’s question would imply the stronger result that \(w\) is primitive if and only if, for some \(n\), \(K_n\) and the one-relator group \(G_n\) defined as in 2 have the same profinite completion.
Indeed, suppose that for some \(n\) the one-relator group \(G_n:=F_k/\langle \langle w^n \rangle \rangle\) has the same profinite completion as the group \(K_n:=F_{k-1}*\mathbb{Z}/n\). Then \(\widehat{G_n}\) is virtually a free profinite group, which would imply by Remeslennikov that \(G_n\) is virtually free. We conclude that \(w\) is primitive by [3].
In the free group \(F_{2k}\) we say that a word \(w\) is a surface word if \(F/\langle \langle w \rangle\rangle\) is isomorphic to the fundamental group of a closed non-positively curved surface. We can similarly recognise surface words:
Corollary 3. \(w\) is a surface word in \(F_{2k}\) if and only if, for all sufficiently large \(n\), the one-relator group \(G_n:=F_{2k}/\langle \langle w^n \rangle \rangle\) is virtually a surface group.
The strategy to prove 1 will be to apply a theorem of Wilton [2] that gives a suitable map of pairs from a surface with boundary to the free group such that the boundary is mapped to the word \(w\) that we are interested in, modify the surface so that there are no accidental parabolics, and then use a result of Agol-Groves-Manning (6 below) to show that the fundamental group of the coned off orbifold injects into the one-relator group \(G_n\) when \(n\) is sufficiently large. We will recall below the relevant notions.
In this section we recall some facts from the theory of Dehn filling that we will need. We refer the reader to [6] and [7] and the references therein for background on relatively hyperbolic groups and Dehn filling.
Let \(G\) be a group which is hyperbolic relative to a finite collection \(\mathcal{P}=\{P_1,\ldots,P_n\}\). Recall that a filling of \(G\) is a choice of subgroups \(N_j \unlhd P_j\), called filling kernels. The quotient by the normal subgroup generated by the \(N_j\) is denoted by \(G(N_1,\dots N_m)\). For any word \(w\) which isn’t a proper power we will view the free group as relatively hyperbolic relative to the subgroup generated by \(\langle w \rangle\) and quotienting out \(\langle \langle w^n \rangle \rangle\) as a filling.
Definition 4. Let \(H\) and \(G\) be relatively hyperbolic groups relative to \(\mathcal{D}=\{D_1, \ldots ,D_m\}\) and \(\mathcal{P}=\{P_1,\ldots,P_n\}\) respectively. A homomorphism \(\phi:H \to G\) is said to respect the peripheral structure (on \(H\)) if, for every \(i\), \(\phi(D_i)\) is conjugate in \(G\) into some \(P_j \in \mathcal{P}\). An element \(h \in H\) is said to be an accidental parabolic of \(\phi\) if it isn’t conjugate in \(H\) into any parabolic subgroup \(D_j^h\) but \(\phi(h)\) is conjugate in \(G\) into some parabolic \(P_i^g\). Hence having no accidental parabolics is when \(\phi(H) \cap P_i^g\) is contained in some \(D_j^h\) for all \(i\) and \(g \in G\).
Definition 5. Suppose \(G\) is a relatively hyperbolic group, relative to \(\mathcal{P}\), and that \(H < G\) is hyperbolic relative to \(\mathcal{D}\) and that the inclusion of \(H\) into \(G\) respects the peripheral structure. A filling \(G \to G(N_1,\dots N_m)\) is an \(H-\)filling if whenever \(H \cap P_i^g\) is nontrivial, \(N_i^ g \subseteq sD_j s^{-1} \subseteq H\) for some \(s\in H\), and \(D_j \in \mathcal{D}\).
The main result we need from the theory of Dehn fillings is
Proposition 6. [7] Let \(H<G\) be a relatively quasi-convex subgroup. For any sufficiently large \(H-\)filling \(G(N_1,\dots N_m)\) of \(G\), the induced map from the induced filling \(H(K_1, \dots K_n)\) into \(G(N_1,\dots N_m)\) is injective.
We will use the following theorem from [2], which is one of the main ingredients used to prove [2] about the existence of surface subgroups in hyperbolic graphs of free groups with cyclic edge groups.
Theorem 7. [2] Let \(F_k\) be a free group and fix \(w\) an imprimitive word. Then there is a compact, connected, hyperbolic surface with boundary \(S\) and an injective map of pairs \((\pi_1(S), \pi_1(\partial S)) \to (F_k, \langle w \rangle)\) that respects the peripheral structure.
Recall that for a finite-index subgroup \(K\) of a group \(G\) and an element \(g \in G\), the degree of \(g\) in \(K\), denoted \(\mathrm{deg}_K (g)\), is the minimal positive integer \(n\) such that \(g^n \in K\).
Definition 8. Let \(\gamma\) be a closed curve on a surface \(\Sigma\) and let \(\Sigma' \to \Sigma\) be a covering map. If \(g \in \pi_1 (\Sigma)\) is freely homotopic to \(\gamma\) and \(n = deg_{\pi_1 (\Sigma' )} (g)\) then \(g^n\) lifts to a curve \(\gamma'\) on \(\Sigma'\). Such a curve \(\gamma'\), defined up to free homotopy on \(\Sigma'\), is called an elevation of \(\gamma\) to \(\Sigma'\).
At the level of fundamental groups, an elevation of \(g \in \pi_1(\Sigma)\) to \(\pi_1(\Sigma')\) is the conjugacy class of a conjugate of a power of \(g\), \(hg^nh^{-1}\) say, which is not a proper power in \(\pi_1(\Sigma')\). \(n\) is the degree of the elevation.
Lemma 9. There exists a compact surface \(\Sigma'\) of negative Euler characteristic such that there is a \(\pi_1\)-injective map of pairs \(f:(\Sigma', \partial\Sigma')\to (\Gamma, \langle {w} \rangle)\) without any accidental parabolics.
Proof. We will modify the surface given by 7 to find a different hyperbolic surface without accidental parabolics.
Suppose that \(\langle g\rangle\) is a cyclic subgroup of \(\pi_1(\Sigma)\) which isn’t conjugate into the boundary but whose image under \(f\) is conjugate in \(F\) to a parabolic, i.e. a subgroup of a conjugate of \(\langle w\rangle\).
By [8] \(g\) must belong to a conjugacy class given by an elevation of \(w\) to \(\pi_1(\Sigma)\). By an argument analogous to that given in [8] there are only finitely many possible conjugacy classes of such \(g\). Choose representatives \(g_1, g_2, \dots, g_n\) for these conjugacy classes. Note that if \(h\) gives rise to an accidental parabolic then it will be a power of some elevation of \(w\).
By results of [9], there exists a finite cover \(\hat{\Sigma}\) of \(\Sigma\) where each conjugacy class is represented by an embedded curve. Up to passing to a further finite cover, we may take the cover to be normal, so all elevations of every conjugacy class are embedded curves in \(\hat{\Sigma}\). Among all possible subsets of curves representing the elevations of the \(g_i\) such that the images of any two curves are disjoint up to isotopy, choose some subset which is maximal under inclusion and cut along the curves in this subset. Since the original surface was hyperbolic, it couldn’t have been an annulus, and in particular one of the connected components \(C\) of the cut surface must be either a surface with genus \(g>0\) or a sphere with \(b\geq 3\) boundary components. We will take \(\Sigma'\) to be \(C\). Note that none of the connected components can contain any accidental parabolics, otherwise we could cut along them as well and this would violate the maximality condition. ◻
Proof of 1. Note that it suffices to show the theorem in the case that \(w\) is not a proper power, so from now we assume this. In [6] it is shown that a hyperbolic group is hyperbolic relative to almost malnormal and quasi-convex subgroups. Since free groups are locally quasi-convex and maximal cyclic subgroups of a non-abelian free group are malnormal, the free group is hyperbolic relative to the subgroup \(\langle w\rangle\), which from now on we take as the only parabolic of \(F_k\). Since \(F_k\) is locally quasi-convex, [10] implies that any finitely generated subgroup of \(F_k\) is relatively quasi-convex for this relatively hyperbolic structure.
Take the surface \(\Sigma'\) from 9. Let \(d\) be the least common multiple of the degrees of the boundary components of \(\Sigma'\). When \(n\) is a multiple of \(d\) the Dehn filling given by the filling kernel \(\langle w^n \rangle\) is an \(H\)-filling. When \(n\) is sufficiently large, 6 implies that the fundamental group \(H\) of the hyperbolic orbifold \(O\) obtained by coning off the boundary components of \(\Sigma'\) by a disc with cone point \(\mathbb{Z}/n\) will inject in the corresponding Dehn filling of \(F\), i.e. \(G_n=F_k/\langle\langle w^n\rangle\rangle\).
Bad 2-orbifolds without boundary, i.e. ones which aren’t finitely covered by a manifold, have been classified [11]. Orbifolds which aren’t bad are said to be good. Since \(\Sigma'\) has either \(b\geq 3\) boundary components or genus \(g>0\), \(O\) isn’t on the list of bad orbifolds, hence is finitely covered by a surface. Following section 2 of [11] we compute that the orbifold Euler characteristic is \(\chi(O) = 2-2g-b(1-\frac{1}{n})\). Elementary algebra shows that this will be negative when \(n>3\). Since \(O\) is good and Euler characteristic is multiplicative under finite covers, \(O\) has a finite cover which is a closed surface of negative Euler characteristic, hence is hyperbolic. The fundamental group of this surface is the desired surface subgroup.
Since it was proved in [12] that \(G_n\) is locally quasi-convex when \(n\geq |w|\), for sufficiently large \(n\) the surface subgroup is quasi-convex. ◻
Remark 10. Since all one-relator groups with torsion are hyperbolic by Newman’s spelling theorem it would be interesting to find surface subgroups in all of them, or at least with sufficiently large torsion. However, it seems difficult to control the degree of the maps on the boundary components of the surface from 7, and in general these will almost certainly have degree \(>1\), so a different strategy will probably be necessary.
Proof of 2. One-relator groups with torsion are hyperbolic and cubulated when \(n\geq 4\) [13], hence virtually special [14] in the sense of [15] and therefore cohomologically separable (also known as ‘good in the sense of Serre’) [16]. If \(w\) isn’t primitive, for large enough \(n\) there exists a quasi-convex surface subgroup \(H\) in the corresponding one-relator group \(G_n\) by 1. Since quasi-convex subgroups of special groups are virtual retracts [15], there exists a finite index subgroup \(G'\) of \(G_n\) which retracts onto \(H\). This implies that \(G'\) has non-vanishing cohomology in degree 2, and since \(G'\) is cohomologically separable its profinite completion \(\widehat{G'}\) does too. This implies \(\widehat{G} \ncong \widehat{K_n}\) since \(\widehat{K_n}\) is virtually a free profinite group. ◻
Remark 11. An analogous proof to that of 1 would show the following generalisation of 1. For any finite set of relators \(r_1, \ldots,r_l\) such that \(\langle x_1, \dots x_k | r_1, \ldots,r_l \rangle\) is irreducible in the sense of [2], there is some \(d\) such that, for all sufficiently large \(n\) which are divisible by \(d\), the quotient \(Q_n:=\langle x_1, \dots x_k | r_1^n, \ldots,r_l^n \rangle\) contains a surface subgroup. For sufficiently large \(n\) the quotient will be small cancellation and hence hyperbolic and cubulated, and furthermore locally quasi-convex by [17]. However, it is no longer possible to deduce primitivity of the relators since, for example, \(\langle a,b |a , bab^{-1}a^{-2}\rangle\) is a presentation of the trivial group but \(bab^{-1}a^{-2}\) isn’t a primitive word.
Proof of 3. If \(w\) is a surface word then \(G_n\) is the fundamental group of the orbifold obtained by taking the corresponding surface with boundary and gluing in a disc with a cone point of order \(n\). This orbifold is good as above, so is virtually a surface. Now suppose \(w\) isn’t a surface word. If \(w\) is primitive, then \(G_n\) is virtually free, hence not virtually a surface. Otherwise, suppose \(w\) is neither primitive nor a surface word. Then the surface subgroup given by 7 is of infinite index. As in the proof of 2, there is a finite index subgroup \(G'\) of \(G_n\) and a retraction \(r: G' \to \pi_1(\Sigma')\) onto a surface subgroup of infinite index. The kernel is infinite, but one-relator groups are virtually torsion-free [18], so the kernel contains an infinite cyclic subgroup. We now conclude verbatim as in the proof of [19]. ◻
The author is greatly indebted to Henry Wilton for suggesting this project and for countless helpful comments on earlier drafts which corrected inaccuracies and substantially improved the quality of exposition. Thanks are also due to Jonathan Fruchter for suggesting 3, his infectious enthusiasm and interest, and suggestions on how to improve the exposition. This work received funding from the European Union (ERC, SATURN, 101076148)