Profinite properties of discrete groups

Abstract

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 §2 we collect some questions that motivated the lectures and this article, and in §3 discuss some examples related to these questions. In §4 we recall profinite groups, profinite completions and the formulation of the questions in the language of the profinite completion. In §5, we recall a particular case of the question of when groups have the same profinite completion, namely Grothendieck's question. In §6 we discuss how the methods of L 2 -cohomology can be brought to bear on the questions in §2, and in §7, we give a similar discussion using the methods of the cohomology of profinite groups. In §8 we discuss the questions in §2 in the context of groups arising naturally in low-dimensional topology and geometry, and in §9 discuss parafree groups. Finally in §10 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. The motivating questionsWe 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 ∈ 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) = 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 setAssuming 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 F n 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(F n ) imply that Γ ∼ = F n ?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 residual...

Around 3-manifold groups

“…A standard way to study residually finite infinite groups is through their finite quotients. The question of how much of the topology of a 3-manifold can be detected from the finite quotients of its fundamental group has attracted some attention recently, see [Re15,Re18]. To a group π we associate the inverse system {π/ K} K where K runs over all finite index normal subgroups of π.…”

“…Following [Re15,Re18] an orientable compact 3-manifold M is called profinitely rigid if π 1 (M) distinguishes π 1 (M) from all other 3-manifold groups. Otherwise it is called profinitely flexible It follows from the geometric decomposition that a compact orientable aspherical 3-manifold which does not contain any essential properly embedded annulus is determined, up to homeomorphism, by its fundamental group, see [Joh79].…”

Profinite groups with NIP theory andp-adic analytic groups

“…By Lemma 4.4, G has finite rank. By Theorem 1.3, we may now assume that the finite index subgroups of G are open, and so are uniformly definable in G. By Corollary 5.4.5 of [20] (extending [26,Theorem 8.4.1]), G has normal subgroups N ≤ A ≤ G such that N is pronilpotent of finite rank, A/N is finitely generated abelian, and G/A is finite. Here, N is the pro-Fitting subgroup of G, that is, the group generated by all the subnormal pro-p subgroups of G (over all primes); it is pronilpotent, so closed.…”

“…By Lemma 4.4, G has finite rank. By Theorem 1.3, we may now assume that the finite index subgroups of G are open, and so are uniformly definable in G. By [20,Corollary 5.4.5] (extending [26,Theorem 8.4.1]), G has normal subgroups N A G such that N is pronilpotent of finite rank, A/N is finitely generated abelian, and G/A is finite.…”

“…Thus, as a full profinite group G 1 is not NIP, or even NTP 2 , by Lemma 4.3. The second is the following 2-generator group mentioned at the end of Chapter 5 of [20]. Let p be a prime, and let (q i : i ∈ N) be a sequence of distinct primes such that p i | q i − 1 for all i.…”