Pro-p subgroups of profinite completions of 3-manifold groups

Wilton, Henry; Zalesskiĭ, Pavel

doi:10.1112/jlms.12067

Cited by 6 publications

(6 citation statements)

References 28 publications

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“…The commensurability class of the geometric 3-manifolds is determined by the profinite completion of the fundamental group of one of its members. The profinite completion of fundamental groups can detect many properties of 3-manifolds as shown in the papers [141], [142], [143] by H. Wilton and P. Zalesskii.…”

Section: Letmentioning

confidence: 99%

Low-dimensional solenoidal manifolds

Verjovsky¹

2022

Preprint

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In this paper we survey n-dimensional solenoidal manifolds for n = 1, 2 and 3, and present new results about them. Solenoidal manifolds of dimension n are metric spaces locally modeled on the product of a Cantor set and an open n-dimensional disk. Therefore, they can be "laminated" (or "foliated") by n-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids i.e., are obtained as the inverse limit of an increasing tower of finite, regular covers of a compact manifold with an infinite and residually finite fundamental group. In this case their structure is very rich since they are principal Cantor-group bundles over a compact manifold and they behave like "laminated" versions of compact manifolds, thus they share many of their properties. These objects codify the commensurability properties of manifolds.To Dennis Sullivan on occasion of his 80th birthday, for all the inspiration and mathematical ideas he has shared with me.

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Section: Letmentioning

confidence: 99%

Low-dimensional solenoidal manifolds

Verjovsky¹

2022

Preprint

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“…Theorem 2.4. [34,Theorem 3.8] Let G be a finitely generated pro-p group acting k-acylindrically on a profinite tree T . Then G is a free pro-p product of vertex stabilizers and a free pro-p group.…”

Section: Pro-p Version Of Itmentioning

confidence: 99%

“…The objective of this paper is to extend results of [33] and [34] to relatively hyperbolic virtually compact special groups and standard arithmetic hyperbolic manifolds of higher dimension.…”

Section: Introductionmentioning

confidence: 99%

“…In [33] it was shown that the geometry type of a geometric 3-manifold is detected by the profinite completion of the fundamental group of the manifold; in particular the profinite version of Hyperbolization theorem and the Seifert conjecture were established there. Moreover, several structure results on profinite completion of 3-manifold and more generally on the profinite completion of virtually compact special hyperbolic groups were proved in [33] and [34].…”

Section: Introductionmentioning

confidence: 99%

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The profinite completion of relatively hyperbolic virtually special groups

Zalesskiĭ¹

2022

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We give a characterization of toral relatively hyperbolic virtually special groups in terms of the profinite completion. We also prove a Tits alternative for subgroups of the profinite completion G of a relatively hyperbolic virtually compact special group G and completely describe finitely generated pro-p subgroups of G. This applies to the profinite completion of the fundamental group of a hyperbolic arithmetic manifold. We deduce that all finitely generated pro-p subgroups of the congruence kernel of a standard arithmetic lattice of SO(n, 1) are free pro-p.

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“…By a combination of the last paragraph of [3] and [17,Theorem 1.3], one obtain the following list of isomorphism types for torsion-free virtually abelian pro-p groups.…”

Section: Polycyclic Pro-p Groupsmentioning

confidence: 99%

Subgroups of pro-$p$ $\mathrm{PD}^3$-groups

Castellano,

Zalesskii

2020

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We study 3-dimensional Poincaré duality pro-p groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-p group G has a nontrivial finitely presented subnormal subgroup of infinite index, then either the subgroup is cyclic and normal, or the subgroup is cyclic and the group is polycyclic, or the subgroup is Demushkin and normal in an open subgroup of G.Also, we describe the centralizers of finitely generated subgroups of 3-dimensional Poincaré duality pro-p groups.

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Pro-p subgroups of profinite completions of 3-manifold groups

Abstract: We completely describe the finitely generated pro-p subgroups of the profinite completion of the fundamental group of an arbitrary 3-manifold. We also prove a pro-p analogue of the main theorem of Bass-Serre theory for finitely generated pro-p groups.

Cited by 6 publications

References 28 publications

Low-dimensional solenoidal manifolds

Low-dimensional solenoidal manifolds

The profinite completion of relatively hyperbolic virtually special groups

Subgroups of pro-$p$ $\mathrm{PD}^3$-groups

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