Volume gradients and homology in towers of residually-free groups

Bridson, Martin R.; Kochloukova, Dessislava H.

doi:10.1007/s00208-016-1387-0

Cited by 13 publications

(34 citation statements)

References 31 publications

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“…Theorem 7.7 can be usefully applied to cases where F is not free; for example, F might be a non-abelian surface group or, more generally, a non-abelian limit group. That these satisfy the condition on the first L 2 -Betti number can be seen in Example 7.2 for surface groups and [20] for limit groups. This leads to an analogue of Theorem 7.1 for paralimit groups.…”

Section: Parafree Groups and Latticesmentioning

confidence: 79%

Nilpotent Completions of Groups, Grothendieck Pairs, and Four Problems of Baumslag

Bridson

Reid

2014

International Mathematics Research Notices

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Abstract. Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology H 2 (−, Z) and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first L 2 -Betti number of a finitely generated parafree group in the same nilpotent genus as a free group of rank r is r − 1. It follows that the reduced C * -algebra of the group is simple if r ≥ 2, and that a version of the Freiheitssatz holds for parafree groups.

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Section: Parafree Groups and Latticesmentioning

confidence: 79%

Nilpotent Completions of Groups, Grothendieck Pairs, and Four Problems of Baumslag

Bridson

Reid

2014

International Mathematics Research Notices

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“…Another standard result about free groups is that if F is a finitely generated free group of rank ≥ 2, then any finitely generated non-trivial normal subgroup of F has finite index (this also holds more generally for Fuchsian groups and limit groups, see [17] for the last statement). As a further corollary of Propositon 6.4 we prove the following.…”

Section: -Betti Numbers and Profinite Completionmentioning

confidence: 99%

Profinite properties of discrete groups

Reid

2015

Groups St Andrews 2013

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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...

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“…The first lemma is an easy consequence of the Bass-Serre theory. For details see [8]. → Q be a short exact sequence of groups.…”

Section: Preliminaries On K(g 1) Cw-complexesmentioning

confidence: 99%

Rank and deficiency gradients of generalized Thompson groups of type F

Kochloukova

2014

Bulletin of the London Mathematical Society

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For an arbitrary sequence (Gs) of subgroups of finite index in the generalized Thompson group Fn,∞, it is shown that the supremum of the minimal number of generators of Gs is finite and the supremum of the absolute value of the deficiency of Gs is finite. In particular, the deficiency gradient of Fn,∞ with respect to (Gs) is 0 provided [G:Gs] tends to infinity. A higher dimensional analogue is considered for n=2.

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Volume gradients and homology in towers of residually-free groups

Cited by 13 publications

References 31 publications

Nilpotent Completions of Groups, Grothendieck Pairs, and Four Problems of Baumslag

Nilpotent Completions of Groups, Grothendieck Pairs, and Four Problems of Baumslag

Profinite properties of discrete groups

Rank and deficiency gradients of generalized Thompson groups of type F

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