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LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART I: GENERAL THEORY

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:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia; colin.d.reid@newcastle.edu.au, george.willis@newcastle.edu.au
GEORGE A. WILLIS
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia; colin.d.reid@newcastle.edu.au, george.willis@newcastle.edu.au

Abstract

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Let $G$ be a totally disconnected, locally compact group. A closed subgroup of $G$ is locally normal if its normalizer is open in $G$. We begin an investigation of the structure of the family of closed locally normal subgroups of $G$. Modulo commensurability, this family forms a modular lattice ${\mathcal{L}}{\mathcal{N}}(G)$, called the structure lattice of $G$. We show that $G$ admits a canonical maximal quotient $H$ for which the quasicentre and the abelian locally normal subgroups are trivial. In this situation ${\mathcal{L}}{\mathcal{N}}(H)$ has a canonical subset called the centralizer lattice, forming a Boolean algebra whose elements correspond to centralizers of locally normal subgroups. If $H$ is second-countable and acts faithfully on its centralizer lattice, we show that the topology of $H$ is determined by its algebraic structure (and thus invariant by every abstract group automorphism), and also that the action on the Stone space of the centralizer lattice is universal for a class of actions on profinite spaces. Most of the material is developed in the more general framework of Hecke pairs.

MSC classification

Primary: 22D05: General properties and structure of locally compact groups
Secondary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e11
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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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