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LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART II: COMPACTLY GENERATED SIMPLE GROUPS

Part of: Structure and classification of infinite or finite groups Locally compact groups and their algebras

Published online by Cambridge University Press:  22 May 2017

PIERRE-EMMANUEL CAPRACE ,
COLIN D. REID  and
GEORGE A. WILLIS
PIERRE-EMMANUEL CAPRACE
Affiliation:
Université catholique de Louvain, IRMP, Chemin du Cyclotron 2, bte L7.01.02, 1348 Louvain-la-Neuve, Belgique; pe.caprace@uclouvain.be
COLIN D. REID
Affiliation:
University of Newcastle, School of Mathematical and Physical Sciences, Callaghan, NSW 2308, Australia; Colin.Reid@newcastle.edu.au, George.Willis@newcastle.edu.au
GEORGE A. WILLIS
Affiliation:
University of Newcastle, School of Mathematical and Physical Sciences, Callaghan, NSW 2308, Australia; Colin.Reid@newcastle.edu.au, George.Willis@newcastle.edu.au

Abstract

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We use the structure lattice, introduced in Part I, to undertake a systematic study of the class $\mathscr{S}$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are nondiscrete. Given $G\in \mathscr{S}$, we show that compact open subgroups of $G$ involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centralizer lattice and local decomposition lattice of $G$ are Boolean algebras. We show that the $G$-action on the Stone space of those Boolean algebras is minimal, strongly proximal, and microsupported. Building upon those results, we obtain partial answers to the following key problems: Are all groups in $\mathscr{S}$ abstractly simple? Can a group in $\mathscr{S}$ be amenable? Can a group in $\mathscr{S}$ be such that the contraction groups of all of its elements are trivial?

MSC classification

Primary: 22D05: General properties and structure of locally compact groups
Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups 20E32: Simple groups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e12
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2017

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