A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable

A PROOF THAT ALL SEIFERT 3-MANIFOLD GROUPS AND ALL VIRTUAL SURFACE GROUPS ARE CONJUGACY SEPARABLE ARMANDO MARTINO Abstract. We prove that the fundamental group of any Seifert 3- manifold is conjugacy separable. That is, conjugates may be distinguished in finite quotients or, equivalently, conjugacy classes are closed in the pro-finite topology. 1. Introduction There has been considerable interest in the separability properties of groups, both from group theorists and topologists. A subset Xof a group Gis called separable if for each g6∈ Xthere exists a map π:G→Q, to a finite group Qsuch that gπ 6∈ Xπ. It is well known that residual finiteness, where X={1}, for a finitely presented group implies a solution to the word problem. Similarly, subgroup separability, where Xis a finitely generated subgroup, implies a solution to the membership problem. And conjugacy separability, where Xis a conjugacy class implies a solution to the conjugacy problem. Additionally, these properties are related to the problem of lifting immersed subspaces of topological spaces to embedded subspaces of finite covers. Recently, [1] have proved that certain Seifert 3-manifold groups are conjugacy separable and in this paper we prove that all Seifert 3-manifold groups are conjugacy separable. They are already known to be residually finite, [5] and subgroup separable, [11]. Our approach is different from that in [1], in that we base our argument on the algebraic structure of a Seifert 3-manifold group as an extension, rather than trying to decompose it into amalgamated free products. One advantage of this approach is that it allows an essentially unified treatment of these groups, although there is a certain distinction which arises depending on whether the Seifert fibred space is ‘built’ from a free group or the fundamental group of a compact surface. Date: April 21, 2005. 1 2 ARMANDO MARTINO Our proof relies on the fact that virtually surface groups are all conjugacy separable. However, while this is already known that virtually free groups are conjugacy separable, it was surprisingly unknown for virtually surface groups. It is important to realise the contrast between conjugacy separability and residual finiteness at this point, since the latter is easily shown to pass to finite extensions whereas for the former this is unclear and possibly untrue. However, it is known that Fuchsian groups are conjugacy separable, and we show how to argue from here to deduce that all virtually surface groups are conjugacy separable. 2. Background 2.1. Pro-finite topology. Given a group, G, the pro-finite topology on G has as a basis all the cosets of finite index subgroups of G. Each such coset is both open and closed in the pro-finite topology. While the separability properties of Gcan be described in terms of the finite quotients of G, it is sometimes more convenient to talk about the pro-finite topology instead. For instance, Gis called residually finite if for any 1 6=g∈G, there exists a finite quotient of Gin which gdoes not map to the identity element. In other words, there is a normal subgroup Nof Gof finite index such that g6∈ N. Equivalently, this means that {1}is closed in the pro-finite topology of G. In fact, this is the same as saying that any one element subset of G is closed or, the seemingly stronger statement, that Gis Hausdorff. The group Gis called subgroup separable if for every finitely generated subgroup, Hof Gand every g∈G−Hthere exists a normal subgroup of finite index, Nof Gsuch that g6∈ HN. Thus, there is a finite quotient of Gin which the images of gand Hare disjoint. As before, this is the same as saying that every finitely generated subgroup is closed. More generally, a subset S⊆Gis called separable if for every g6∈ S, there is a finite quotient of Gin which gand Shave disjoint images. Equivalently, Sis separable if it is closed. A group Gis called conjugacy separable if conjugacy classes are separable. If we have a subgroup Hof G, then there are two possible topologies one can put on H. Namely, the subspace topology and the pro-finite topology of Hitself. In general the subspace topology may be more coarse, but not if Hhas finite index. Lemma 2.1. Let Hbe a finite index subgroup of G. Then the subspace topology and the pro-finite topology of Hare the same. CONJUGACY SEPARABILITY OF SEIFERT MANIFOLDS 3 Proof. If Kis a finite index subgroup of Gthen Kx ∩His either empty, or is a coset of K∩H, which has finite index in H. Hence the subspace topology is contained in the pro-finite topology. Conversely, for any K0which is a finite index subgroup of H,K0xis open in G, since K0has finite index in G. Hence the pro-finite topology is contained in the subspace topology. 2 2.2. Seifert fibred spaces. Throughout the paper we shall use the term surface group to mean a group isomorphic to the fundamental group of a compact surface. This includes the free abelian group of rank 2, the closed hyperbolic surface groups as well as any finitely generated free group. Since we are generally working up to finite index, we will mainly restrict our attention to the orientable case. Using standard terminology, we will call a group a virtual surface group if it has a surface subgroup of finite index. A Seifert fibred space is a 3-manifold that is ‘almost’ a bundle. This property can be made more precise when one considers the fundamental group, π1(M) of a Seifert fibred space, M. Theorem 2.2. Let Mbe a Seifert fibred space. Then there exists a short exact sequence, 1→C→π1(M)→H such that Cis cyclic and Hcontains a finite index subgroup isomorphic to a surface group. Proof. This is proved in [12], where it is shown that His the fundamental group of a 2-orbifold. By [12], such a group is virtually a surface group. We note that it is sufficient to consider the case where the 2-orbifold is without boundary, by the remarks in [12], and since finitely generated subgroups of surface groups are surface groups (recall that we are including free groups as surface groups). In fact, it turns out that if π1(M) is infinite - the only real case of interest for us - then Cis also infinite. But we shall not need to use this fact. 2 There is a rich literature dealing with the subject of Seifert fibred spaces ([6], [14], for instance) and their role in the theory of 3-manifolds. For our purposes, however, the Theorem above provides sufficient information on their fundamental groups in order to demonstrate conjugacy separability. 3. Virtual Surface Groups The goal of this section is to prove that any finitely generated group having a surface subgroup of finite index is conjugacy separable. Since virtually free and virtually free abelian groups are already known to be conjugacy separable, we shall in fact only be considering the case of closed 4 ARMANDO MARTINO hyperbolic surface groups. In fact, since in [3] it has been proved that Fuchsian groups are conjugacy separable, we shall show that this is sufficient for one to deduce that any finite extension of a closed hyperbolic surface group is conjugacy separable. The connection between the general case and the Fuchsian case is provided by the Nielsen Realisation Theorem. First, however, we start with some terminology. Definition 3.1. Given a group Swe call an automorphism, φ, of Svirtually inner if φnis inner for some integer n. Given a virtually inner automorphism, φ, let S∗φdenote the group hS, t :tn=x, t−1gt =gφ, for all g∈Si, where nis chosen to be the least integer such that φnis inner and xis chosen so that gφn=x−1gx for all g∈S. This group has Sas a subgroup of finite index. Also, given an automorphism φof S, we consider the equivalence relation, ∼φon Swhere g1∼φg2if and only if there exists an h∈Ssuch that (h−1φ)g1h=g2. We call this relation ”twisted-φconjugacy” with ”twistedφconjugacy classes”. We denote the twisted-φconjugacy class of an element hby [h]φ. The main results we shall use in the section are the following, Theorem 3.2 (Nielsen Realisation Theorem, [8], [10]).Let Sbe the fundamental group of a closed hyperbolic surface. Let Hbe a finite subgroup of Out(S)and let Gbe the pre-image of Hin Aut(S). Then Ghas Inn(S)∼ =S as a subgroup of finite index and is a Fuchsian group. Theorem 3.3 ([3]).Fuchsian groups are conjugacy separable. Putting the above two together immediately produces the following crucial corollary. Corollary 3.4. Let Sbe the fundamental group of a closed hyperbolic surface, and φa virtually inner automorphism of S. Then S∗φis conjugacy separable. Proof. Clearly, the image of φin Out(S) is a finite cyclic subgroup whose pre-image in Aut(S) is isomorphic to S∗φ.2 The purpose of introducing our seemingly cumbersome terminology (twisted-φconjugacy) is that it enables us to pass between finite extensions of a group, S, by describing the separability properties of the extension solely in terms of S. Proposition 3.5. Let Sbe a conjugacy separable group and φa virtually inner automorphism of S. If S∗φis conjugacy separable, then twisted-φ conjugacy classes in Sare closed in the pro-finite topology of S. CONJUGACY SEPARABILITY OF SEIFERT MANIFOLDS 5 Proof. Note that Sis a finite index subgroup of S∗φ, so a subset of S is closed in the pro-finite topology of Sif and only if it is closed in the pro-finite topology of S∗φ. Now consider the element, tg, for g∈S. Clearly, every conjugate of tg is a conjugate of tg by some element of S. Moreover, if x∈Sthen, x−1tgx =t(x−1φ)gx. Hence the conjugacy class of tg in S∗φis equal to t([g]φ), where the twisted-φconjugacy class is understood to be in S. Now, since group multiplication is a homeomorphism, [g]φmust be closed in S∗φand hence in S.2 Proposition 3.6. Let Gbe a group which has Sas a normal subgroup of finite index. Suppose that all twisted-φconjugacy classes of Sare closed in the pro-finite topology of S, for all virtually inner automorphisms φ. Then Gis conjugacy separable. Proof. Again, as Shas finite index, a subset of Sis closed in Sif and only if it is closed in G. Now consider a g∈G. Let x1, . . . , xrbe a set of coset representatives for Sin Gand let gi=x−1gxi. Let φibe the automorphism of Sinduced by conjugation by the element gi. Clearly, each φiis virtually inner. We observe the following, {w−1gw :w∈G}=Sr i=1{x−1gix:x∈S} =Sr i=1 gi[1]φi, where [1]φiis understood to be a twisted-φiconjugacy class in S. By hypothesis, each [1]φiis closed in Sand thence in G. Thus each gi[1]φiis closed in Gand thus so is the (finite) union. Thus we have shown that every conjugacy class in Gis closed, which is another way of saying that G is conjugacy separable. 2 It is now clear how to put these results together to get, Theorem 3.7. Let Gbe a group having a surface subgroup of finite index. Then Gis conjugacy separable. Proof. Let Sbe a surface subgroup of finite index which we assume, without loss of generality, is normal in G. If Sis free this is proved by [2]. If Sis free abelian (the torus case), this is proved by [4]. So we are left with the case where Sis the fundamental group of a closed hyperbolic surface. Proposition 3.5 and Corollary 3.4 imply that all twistedφconjugacy classes of Sare closed, for any virtually inner φ. Proposition 3.6 then implies that Gis conjugacy separable. 2 6 ARMANDO MARTINO 4. Extensions The main goal of this section is to prove Theorem 4.3, which says that finite-by-surface groups are surface-by-finite. That is, they are virtually surface groups. We note that this is already known, see [7], exercise 4.7, but we felt that since it seems to be a key observation which allows us to construct our proof of the fact that Seifert 3-manifold groups are conjugacy separable, it would be beneficial to include a proof of it. Additionally, the proof we present here is, as far as we are aware, new and is elementary up to some well known results concerning the residual properties of free groups and nilpotent groups. We start with the following lemma. Lemma 4.1. Let Fbe a finitely generated free group and 16=g∈F. For any integer nthere exists a finite quotient, Qnof F, such that the image of gis a central element of order nin Qn. Proof. It is clearly sufficient to prove the Lemma in the case where nis a power of some prime. So we consider an arbitrary prime pand will show that there is a quotient of Fin which the image of gis central and has order pkfor any natural number k. Now consider the central series for F. Write γ0=Fand γi+1 = [F, γi]. As Fis residually nilpotent ([9]) we can find a least isuch that g6∈ γi. Hence gis non-trivial and central in the free nilpotent group F/γi. The group F/γiis a torsion free nilpotent group, and hence is residually a finite p-group for all primes p([13] or deduce this from [9] again). Thus we can find a sequence of quotients Qkof Fsuch that each Qkis a p-group in which the image of gis central and ghas order pnkfor some unbounded sequence nk. (Specifically, Qkis a finite p-group quotient of F/γiin which the images of gpi, for 1 ≤i≤k, are non-trivial.) However, we note that if his a central element of a finite p-group Qwith order pr, then there is a quotient of Qin which his central and has order pr−1. Namely, quotient out by all central elements of order p. Hence we are done. 2 Lemma 4.2. Consider the short exact sequence of groups, 1→G→Γ→πH→1 where Gis a finite subgroup of the centre of Γand His a surface group. Then Γhas a finite index subgroup Γ0such that Γ0∩Gis trivial. Proof. This is trivial if His free so we assume that His a one-relator group and we write H=hX:ri. Now let Fbe the free group on the set X, and ρ:F→Hthe natural epimorphism. By the universal property of CONJUGACY SEPARABILITY OF SEIFERT MANIFOLDS 7 free groups, there is a map σ:F→Γ and a commuting triangle, F σ ÄÄÄ Ä Ä Ä Ä Ä Ä Äρ ÃÃ @ @ @ @ @ @ @ @ Γπ//H Now rσ is clearly in the kernel of πand hence is an element of G. Suppose the order of rσ is n. Then, by Lemma 4.1, there exists a finite quotient, Q, of Fin which the image of ris central and has order n. Let F0be the kernel of the map from Fto Qand consider the subgroup Γ0=F0σ. Clearly, Γ0 is a finite index subgroup of Γ. We claim that Γ0is the required subgroup. For consider an element w∈Γ0∩G. Then w=uσ for some u∈F0. Moreover, as wlies in the kernel of π,umust lie in the kernel of ρ. Hence we can write uas a product of conjugates of r±1as follows, u= k Y i=1 g−1 ir²igi where each ²i=±1 and the gi∈F. Note that since ris central in F/F0we have that u=rmin F/F0, where m=Pk i=1 ²i. In fact, since rhas order exactly nin F/F0and u∈F0we also have that mis a multiple of n. However, since rσ is central in Γ we also get that w=uσ = (rσ)m. As rσ has order nand mis a multiple of n, we deduce that Γ0∩Gis trivial. 2 Theorem 4.3. Consider a short exact sequence of groups, 1→G→Γ→H→1, where Gis a finite group and His a surface group. Then Γhas a subgroup of finite index which is a surface group. In particular, Γis conjugacy separable. Proof. Note that Γ acts on Gby conjugation. As this is a finite group, the kernel of this action, Γ1is a finite index subgroup of Γ which centralises G. In particular, G∩Γ1is a finite central subgroup of Γ1. Since the image of Γ1in His a finite index subgroup of Hand hence a surface group, we may apply Lemma 4.2 to deduce that Γ1and hence Γ has a subgroup of finite index which intersects Gtrivially. Since this subgroup is isomorphic to a subgroup of finite index in H, we have shown that Γ is virtually a surface group. Hence, by Theorem 3.7, Γ is conjugacy separable. 2 5. Main argument Given a Seifert 3-manifold, M, we will show that π1(M) is conjugacy separable by considering the short exact sequence given by Theorem 2.2. 1→C→π1(M)→H→1, 8 ARMANDO MARTINO where Cis cyclic and His virtually a surface group. In particular, we do this by analysing the action on the normal subgroup C. In fact, by Theorem 4.3, it is sufficient to consider the case when Cis infinite cyclic, so henceforth we shall assume that C∼ =Z. We let hdenote the generator of this Zand note that every conjugate of his equal to either hor h−1. In particular, the subgroup hhkiis a normal subgroup of π1(M) for each integer k. Our strategy will be to show that if two elements in π1(M) are not conjugate, then they will also fail to be conjugate in a quotient of π1(M) by some hhki. Since these quotients are all conjugacy separable by the previous section, this is clearly sufficient. The crux of the argument that follows is the content of the following lemma. Lemma 5.1. Let g∈π1(M). Then there exist integers λ, λ0so that g conjugate to ghnin π1(M)if and only if n=kλ or n=kλ +λ0, for some integer k. Proof. Let Cbe the pre-image in π1(M) of the centraliser of gin π1(X). That is, Cis precisely the subgroup of elements which conjugate gto an element of the form ghnfor some integer n. Let C+be the subgroup of C of index at most 2 which centralise h. Consider x1, x2∈C+and suppose that x−1 igxi=ghni. Then it is easy to check that x∓1 2x−1 1gx1x±1 2=ghn1±n2. In other words, the set {hn:x−1gx =ghnfor some x∈C+}is a subgroup of hhi. Thus it has a generator hλand we have shown that ghn=x−1gx for some x∈C+if and only if n=kλ. If C=C+we are done by setting λ0= 0. Otherwise we can find a y∈C−C+and an integer λ0such that y−1gy =ghλ0. The conclusion of this lemma is now immediate on noting that, y−1ghkλy=gh−kλ+λ0, and that C=C+∪C+y.2 Theorem 5.2. All Seifert 3-manifold groups are conjugacy separable. Proof. By Theorem 4.3, it is sufficient to prove the theorem in the case where we have a short exact sequence of groups associated to a Seifert 3- manifold, M, of the form 1→Z→π1(M)→H→1, where His a virtual surface group and Zis generated by an element h. So let us consider two elements g, g1∈π1(M) which are not conjugate. In order to show that these elements are not conjugate in some finite quotient, CONJUGACY SEPARABILITY OF SEIFERT MANIFOLDS 9 it is clearly sufficient to show that they are not conjugate in some conjugacy separable quotient of π1(M). Thus we may assume that the images of gand g1are conjugate in H, since His a conjugacy separable quotient of π1(M), by Theorem 3.7. After taking suitable conjugates, we may further assume that gand g1have the same image in H. Thus g1=ghnfor some integer n. However, by Lemma 5.1, there exist integers λ, λ0so that an element of the form ghmis conjugate to gin π1(M) if and only if m=kλ or m=kλ +λ0for some integer k. In particular, since gand g1are not conjugate in π1(M), neither can they be conjugate in the quotient π1(M)/hhλi. However, again by Theorem 4.3, π1(M)/hhλi is conjugacy separable and we are done. 2 Acknowledgments We gratefully acknowledge the support of the CRM and its staff and thank evryone at the CRM for their hospitality. 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