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Profinite Properties of Discrete Groups

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Profinite Properties of Discrete Groups PROFINITE PROPERTIES OF DISCRETE GROUPS ALAN W. REID 1. Introduction This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In x2 we collect some questions that motivated the lectures and this article, and in x3 discuss some examples related to these questions. In x4 we recall profinite groups, profinite completions and the formulation of the questions in the language of the profinite completion. In x5, we recall a particular case of the question of when groups have the same profinite completion, namely Grothendeick's question. In x6 we discuss how the methods of L2-cohomology can be brought to bear on the questions in x2, and in x7, we give a similar discussion using the methods of the cohomology of profinite groups. In x8 we discuss the questions in x2 in the context of groups arising naturally in low-dimensional topology and geometry, and in x9 discuss parafree groups. Finally in x10 we collect a list of open problems that may be of interest. Acknoweldgement: The material in this paper is based largely on joint work with M. R. Bridson, and with M. R. Bridson and M. Conder and I would like to thank them for their collaborations. I would also like to thank the organizers of Groups St Andrews 2013 for their invitation to deliver the lectures, for their hopsitality at the conference, and for their patience whilst this article was completed. 2. The motivating questions We begin by recalling some terminology. A group Γ is said to be residually finite (resp. residually, nilpotent, residually-p, residually torsion-free-nilpotent) if for each non-trivial γ 2 Γ there exists a finite group (resp. nilpotent group, p-group, torsion-free-nilpotent group) Q and a homomorphism φ :Γ ! Q with φ(γ) 6= 1. 2.1. If a finitely-generated group Γ is residually finite, then one can recover any finite portion of its Cayley graph by examining the finite quotients of the group. It is therefore natural to wonder whether, under reasonable hypotheses, the set C(Γ) = fG : G is a finite quotient of Γg might determine Γ up to isomorphism. Assuming that the groups considered are residually finite is a natural condition to impose, since, first, this guarantees a rich supply of finite quotients, and secondly, one can always form the free product Γ∗S where S is a finitely generated infinite simple group, and then, clearly C(Γ) = C(Γ∗S). Henceforth, unless otherwise stated, all groups considered will be residually finite. The basic motivating question of this work is the following due to Remesselenikov: Question 1: If Fn is the free group of rank n, and Γ is a finitely-generated, residually finite group, supported in part by NSF grants. 1 2 ALAN W. REID ∼ then does C(Γ) = C(Fn) imply that Γ = Fn? This remains open at present, although in this paper we describe progress on this question, as well as providing structural results about such a group Γ (should it exist) as in Question 1. Following [31], we define the genus of a finitely generated residually finite group Γ to be: G(Γ) = f∆ : C(∆) = C(Γ)g: This definition is taken, by analogy with the theory of quadratic forms over Z where two integral quadratic forms can be locally equivalent (i.e. at all places of Q), but not globally equivalent over Z. Question 2: Which finitely generated (resp. finitely presented) groups Γ have G(Γ) = fΓg? Question 3: Which finitely generated (resp. finitely presented) groups Γ have jG(Γ)j > 1? Question 4: How large can jG(Γ)j be for finitely generated (resp. finitely presented) groups? Question 5: What group theoretic properties are shared by (resp. are different for) groups in the same genus? In addition, if P is a class of groups, then we define G(Γ; P) = f∆ 2 P : C(∆) = C(Γ)g; and can ask the same questions upon restricting to groups in P. 2.2. Rather than restricting the class of groups in a genus, we can ask to distinguish finitely gener- ated groups by restricting the quotient groups considered. A particularly interesting case of this is T the following. Note first that, a group Γ is residually nilpotent if and only if Γn = 1, where Γn, the n-th term of the lower central series of Γ, defined inductively by setting Γ1 = Γ and defining Γn+1 = h[x; y]: x 2 Γn; y 2 Γi: Two residually nilpotent groups Γ and Λ are said to have the same nilpotent genus if they have the ∼ same lower central series quotients; i.e. Γ=Γc = Λ=Λc for all c ≥ 1. Residually nilpotent groups with the same nilpotent genus as a free group are termed parafree. In [10] Gilbert Baumslag surveyed the state of the art concerning groups of the same nilpotent genus with particular emphasis on the nature of parafree groups. We will discuss this in more detail in x9 below. 3. Some examples We begin with a series of examples where one can something about Questions 1{4. 3.1. We first prove the following elementary result. Proposition 3.1. Let Γ be a finitely generated Abelian group, then G(Γ) = fΓg. Proof. Suppose first that ∆ 2 G(Γ) and ∆ is non-abelian. We may therefore find a commutator c = [a; b] that is non-trivial. Since ∆ is residually finite there is a homomorphism φ : ∆ ! Q, with Q finite and φ(c) 6= 1. However, ∆ 2 G(Γ) and so Q is abelian. Hence φ(c) = 1, a contradiction. ∼ r ∼ s Thus ∆ is Abelian. We can assume that Γ = Z ⊕ T1 and ∆ = Z ⊕ T2, where Ti (i = 1; 2) are finite Abelian groups. It is easy to see that r = s, for if r > s say, we can choose a large prime p such r that p does not divide jT1jjT2j, and construct a finite quotient (Z=pZ) that cannot be a quotient of ∆. In addition if T1 is not isomorphic to T2, then some invariant factor appears in T1 say, but not in T2. One can then construct a finite abelian group that is a quotient of T1 (and hence Γ1) but not of Γ2. tu Note that the proof of Proposition 3.1 also proves the following. PROFINITE PROPERTIES OF DISCRETE GROUPS 3 Proposition 3.2. Let Γ be a finitely generated group, and suppose that ∆ 2 G(Γ). Then Γab =∼ ∆ab. In particular b1(Γ) = b1(∆). 3.2. Remarkably, moving only slightly beyond Z to groups that are virtually Z, the situation is dramatically different. The following result is due to Baumslag [9]. We include a sketch of the proof. Theorem 3.3. There exists non-isomorphic meta-cyclic groups Γ1 and Γ2 for which C(Γ1) = C(Γ2). Indeed, both of these groups are virtually Z and defined as extensions of a fixed finite cyclic group F by Z. Sketch Proof: What Baumslag actually proves in [9] is the following, and this is what we sketch a proof of: (*) Let F be a finite cyclic group with an automorphism of order n, where n is different from 1, 2, 3, 4 and 6. Then there are at least two non-isomorphic cyclic extensions of F , say Γ1 and Γ2 with C(Γ1) = C(Γ2). Recall that the automorphism group of a finite cyclic group of order m is an abelian group of order φ(m). So in (*) we could take F to be a cyclic group of order 11, which has an automorphism of order 5. Now let F =< a > be a cyclic group of order m, and as in (*) assume that it admits an automorphism α of order n as in (*). Assume that α(a) = ar. Now some elementary number theory (using that φ(m) > 2 by assumption) shows that we can find an integer ` such that (`; n) = 1, and (i) α` 6= α; and (ii) α` 6= α−1: m −1 r Now define Γ1 =< a; bja = 1; b ab = a > to be the split extension of F induced by α and m −1 r` ` Γ2 =< a; cja = 1; c ac = a > be the split extension of F induced by α . The key claims to be established are that Γ1 and Γ2 are non-isomorphic, and that they have the same genus. That the groups are non-isomorphic can be checked directly as follows. If θ :Γ1 ! Γ2 is an isomorphism, then θ must map the set of elements of finite order in Γ1 to those in Γ2; that is to say θ preserves F , and so induces an automorphism of F . Thus θ(a) = as where (s; m) = 1. Moreover ∼ t since the quotients Γi=F = Z for i = 1; 2, it follows that θ(b) = c a where = ±1 and t is an integer. Now consider θ(ar). When θ(b) = cat we get: α(as) = ars = θ(ar) = θ(bab−1) = (cat)as(cat)−1 = α`(as); and it follows that α = α`. A similar argument holds when θ(b) = c−1at to show α−1 = α`, both of which are contradictions to (ii) above. We now discuss proving that the groups are in the same genus. Setting P = Γ1 × Z, Baumslag [9] shows that P is isomorphic to Γ2 × Z. That Γ1 and Γ2 have the same genus now follows from a result of Hirshon [34] (see also [9]) where it is shown that (see Theorem 9 of [34]): Proposition 3.4.
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