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On the residual and profinite closures of commensurated subgroups

On the residual and profinite closures of commensurated subgroups
On the residual and profinite closures of commensurated subgroups
The residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtu- ally normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.
profinite closure, reisidual finiteness
0305-0041
411-432
Caprace, Pierre-Emmanuel
a89ce108-f919-4fbd-9d6f-99b9af6abb81
Kropholler, Peter H.
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Reid, Colin D.
e6ef4078-a882-4960-8c25-fd97d1679b1a
Wesolek, Phillip
726315f9-6185-4951-a47b-26d04e0f3cd8
Caprace, Pierre-Emmanuel
a89ce108-f919-4fbd-9d6f-99b9af6abb81
Kropholler, Peter H.
0a2b4a66-9f0d-4c52-8541-3e4b2214b9f4
Reid, Colin D.
e6ef4078-a882-4960-8c25-fd97d1679b1a
Wesolek, Phillip
726315f9-6185-4951-a47b-26d04e0f3cd8

Caprace, Pierre-Emmanuel, Kropholler, Peter H., Reid, Colin D. and Wesolek, Phillip (2020) On the residual and profinite closures of commensurated subgroups. Mathematical Proceedings of the Cambridge Philosophical Society, 169 (2), 411-432. (doi:10.1017/S0305004119000264).

Record type: Article

Abstract

The residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtu- ally normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

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On the residual and profinite closures of commensurated subgroupspdf - Accepted Manuscript
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Accepted/In Press date: 14 May 2019
e-pub ahead of print date: 30 July 2019
Published date: September 2020
Keywords: profinite closure, reisidual finiteness

Identifiers

Local EPrints ID: 431925
URI: http://eprints.soton.ac.uk/id/eprint/431925
ISSN: 0305-0041
PURE UUID: ddc9687f-77a6-4a0f-83dc-4b18cd535eb4
ORCID for Peter H. Kropholler: ORCID iD orcid.org/0000-0001-5460-1512

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Date deposited: 21 Jun 2019 16:30
Last modified: 11 Mar 2022 05:02

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Contributors

Author: Pierre-Emmanuel Caprace
Author: Colin D. Reid
Author: Phillip Wesolek

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