Virtual pro-<i>p</i> properties of 3-manifold groups

Wilkes, Gareth

doi:10.1515/jgth-2016-0060

Cited by 14 publications

(12 citation statements)

References 42 publications

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“…In particular, determines whether or not is irreducible. While this work was in progress we discovered that a similar result has also been proved in the pro- setting by Wilkes, using -cohomology [Wil17a, Proposition 6.2.4]. Our proof is different, using the continuous cohomology of the profinite completion, and naturally generalizes to our next theorem, which shows that the profinite completion determines the JSJ decomposition of .…”

supporting

confidence: 65%

“…As a warm-up, we show that the profinite completion of a -manifold group determines its Kneser–Milnor decomposition. As noted above, this result can also be obtained using methods from -cohomology [Wil17a]. Recall that a closed -manifold is irreducible if every embedded -sphere bounds a -ball; equivalently, does not admit a non-trivial splitting over the trivial subgroup.…”

Section: The Kneser–milnor Decompositionmentioning

confidence: 99%

“…See Theorem 4.3 for a more precise statement, phrased in terms of profinite Bass–Serre trees . Partial results along the lines of Theorem B have also been obtained by Wilkes [Wil17a, Theorem I].…”

mentioning

confidence: 94%

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Profinite detection of 3-manifold decompositions

Wilton¹,

Zalesskiĭ²

2019

Compositio Math.

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The profinite completion of the fundamental group of a closed, orientable 3-manifold determines the Kneser-Milnor decomposition. If M is irreducible, then the profinite completion determines the Jaco-Shalen-Johannson decomposition of M .When trying to distinguish two compact 3-manifolds M, N, in practice the easiest method is often to compute some finite quotients of their fundamental groups, and notice that there is a finite group Q which is a quotient of π 1 M, say, but not of π 1 N. It would be very useful, both theoretically and in practice, to know that this method always works. The set of finite quotients of a group Γ is encoded by the profinite completionΓ (the inverse limit of the system of finite quotient groups), and so one is naturally led to the following question.Question 0.1. Let M be a compact, orientable 3-manifold. To what extent is π 1 M determined by its profinite completion?In particular, if M is determined among all compact, orientable 3-manifolds by π 1 M, then M is said to be profinitely rigid. If there are at most finitely many compact, orientable 3-manifolds N with π 1 M ≅ π 1 N, then M is said

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supporting

confidence: 65%

Section: The Kneser–milnor Decompositionmentioning

confidence: 99%

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Profinite detection of 3-manifold decompositions

Wilton¹,

Zalesskiĭ²

2019

Compositio Math.

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“…Residual properties of graphs of groups have long been a subject of study and have, for instance, been particularly important in relation to 3-manifold groups [1,3,7,8]. Any study of such properties almost inevitably involves a reduction to the study of graphs of finite groups.…”

Section: Introductionmentioning

confidence: 99%

A criterion for residual 𝑝-finiteness of arbitrary graphs of finite 𝑝-groups

Wilkes¹

2019

Journal of Group Theory

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“…Residual properties of graphs of groups have long been a subject of study, and have for instance been particularly important in relation to 3-manifold groups [Hem87,WZ10,AF13,Wil17]. Any study of such properties almost inevitably involves a reduction to the study of graphs of finite groups.…”

Section: Introductionmentioning

confidence: 99%