Grothendieck’s problem for 3-manifold groups

Long, D. D.; Reid, Alan W.

doi:10.4171/ggd/135

Cited by 20 publications

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“…There are many classes of groups Γ that can never have a subgroup P for which (Γ, P ) is a Grothendieck Pair; as in [40], we call such groups Grothendieck Rigid.…”

Section: 2mentioning

confidence: 99%

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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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“…There are many classes of groups Γ that can never have a subgroup P for which (Γ, P ) is a Grothendieck Pair; as in [40], we call such groups Grothendieck Rigid.…”

Section: 2mentioning

confidence: 99%

“…For those modelled on SOL geometry, separability of subgroups can be established directly and the result follows. ⊔ ⊓ Remark: The case of finite co-volume Kleinian groups was proved in [40] without using the LERF assumption. Instead, character variety techniques were employed.…”

Section: 2mentioning

confidence: 99%

Profinite properties of discrete groups

Reid

2015

Groups St Andrews 2013

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“…(A standard argument shows that two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic.) Bridson, Conder and Reid have answered the corresponding question for Fuchsian groups positively [BCR14], while Long and Reid have given a positive answer to a related question [LR11]. We refer the reader to Section 8 of [Rei13] for a discussion of this and related problems.…”

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confidence: 99%

Distinguishing geometries using finite quotients

Wilton

Zalesskiĭ

2017

Geom. Topol.

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We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to Z 2 . This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for which the profinite completion of the fundamental group has a non-trivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-p subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-p.In a heuristic commonly used to distinguish 3-manifolds M and N, one computes all covers M 1 , . . . , M m of M and N 1 , . . . , N n of N up to some small degree, and then compares the resulting finite lists (M i ) and (N j ). If they can be distinguished, then one has a proof that M and N were not homeomorphic.It is natural to ask whether this method always works. A more precise question was formulated by Calegari-Freedman-Walker in [CFW10], who asked whether the fundamental group of a 3-manifold is determined by its finite quotients, or equivalently, by its profinite completion. (A standard argument shows that two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. In fact, the Geometrization theorem is equivalent to the Hyperbolization theorem and the Seifert conjecture, together with the Elliptization theorem, which asserts that M is spherical if and only if π 1 M is finite [Sco83]. Since 3-manifold groups are residually finite [Hem87], π 1 M is finite if and only if its profinite completion is, so there is no distinct profinite analogue of the Elliptization theorem. Theorems A and B can therefore be thought of as providing 2 a complete profinite analogue of the Geometrization theorem (although one should note, of course, that our proofs rely essentially on Geometrization). In Theorem 8.4, we proceed to show that the profinite completion of the fundamental group detects the geometry of a closed, orientable, irreducible 3-manifold.Alternatively, Theorem A can be thought of as a classification result for the abelian subgroups of profinite completions of fundamental groups of hyperbolic manifolds. In fact, we prove a much more general 'Tits alternative' for profinite completions of 3-manifold groups. We use the notation Z π to denote p∈π Z p .Theorem C. If M is any compact 3-manifold and H is a closed subgroup of π 1 M that does not contain a free non-abelian pro-p subgroup for any prime p then H is on the following list:1. H is conjugate to the completion of a virtually soluble subgroup of π 1 M; or, where π and σ a...

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“…The number of homeomorphism classes in X T V (M ) of hyperbolic fibered 3-manifolds N with fiber of genus g ≥ 1 is finite for every M . Is this number unbounded, when M runs over the set of hyperbolic fibered 3-manifolds with fiber of given genus?The pairs of manifolds from Theorem 1.1 and Corollary 1.3 also give a negative answer to a question stated by Long and Reid in [36] (see also Remark 3.7 in [8]), as follows:Corollary 1.4. For any m ≥ 2 there exist torus bundles whose fundamental groups have isomorphic profinite completions although they are pairwise not isomorphic.…”

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confidence: 99%

Torus bundles not distinguished by TQFT invariants

Funar

2013

Geom. Topol.

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Abstract. We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in SL(2, Z) and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementary study of a family of Pell equations. A key ingredient is the congruence subgroup property of modular representations, as it was established by Coste and Gannon, Bantay, Xu for various versions of TQFT, and lastly by Ng and Schauenburg for the Drinfeld doubles of spherical fusion categories. On the other side we prove that two torus bundles over the circle with the same quantum invariants are (strongly) commensurable. The examples above show that this is the best that it could be expected.

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Grothendieck’s problem for 3-manifold groups

Abstract: Abstract. The following problem was posed by Grothendieck:Let u W H

Cited by 20 publications

References 46 publications

Profinite properties of discrete groups

Profinite properties of discrete groups

Distinguishing geometries using finite quotients

Torus bundles not distinguished by TQFT invariants

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