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On the residual and profinite closures of commensurated subgroups

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  30 July 2019

PIERRE–EMMANUEL CAPRACE ,
PETER H. KROPHOLLER ,
COLIN D. REID  and
PHILLIP WESOLEK
PIERRE–EMMANUEL CAPRACE
Affiliation:
Université Catholique de Louvain, IRMP, Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgique. e-mail: pe.caprace@uclouvain.be
PETER H. KROPHOLLER
Affiliation:
Mathematical Sciences, University of Southampton. e-mail: p.h.kropholler@soton.ac.uk
COLIN D. REID
Affiliation:
University of Newcastle, School of Mathematical and Physical Sciences, Callaghan, NSW 2308, Australia. e-mail: colin.d.reid@newcastle.edu.au
PHILLIP WESOLEK
Affiliation:
Binghamton University, Department of Mathematical Sciences, PO Box 6000, Binghamton, New York 13902-6000, U.S.A. e-mail: pwesolek@binghamton.edu
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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 virtually 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.

MSC classification

Primary: 20E26: Residual properties and generalizations; residually finite groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 411 - 432
Copyright
© Cambridge Philosophical Society 2019

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Footnotes

†

F.R.S.-FNRS senior research associate, supported in part by EPSRC grant no EP/K032208/1.

‡

supported by EPSRC grants no EP/K032208/1 and EP/ N007328/1.

§

ARC DECRA fellow, supported in part by ARC Discovery Project DP120100996.

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