- Title
- Locally normal subgroups of totally disconnected groups. Part I: general theory
- Creator
- Caprace, Pierre-Emmanuel; Reid, Colin D.; Willis, George A.
- Relation
- ARC.DP0984342 | ARC|DP120100996 http://purl.org/au-research/grants/arc/DP120100996
- Relation
- Forum of Mathematics, Sigma Vol. 5, no. e11
- Publisher Link
- http://dx.doi.org/10.1017/fms.2017.9
- Publisher
- Cambridge University Press
- Resource Type
- journal article
- Date
- 2017
- Description
- 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 LN(G), called the structure lattice of G. We show that admits a canonical maximal quotient H for which the quasicentre and the abelian locally normal subgroups are trivial. In this situation LN(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 is H 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.
- Subject
- locally compact groups; closed subgroups; structure lattice; centralizer lattice; Boolean algebra; Hecke pairs
- Identifier
- http://hdl.handle.net/1959.13/1398895
- Identifier
- uon:34494
- Identifier
- ISSN:2050-5094
- Rights
- © The Author(s) 2017. 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.
- Language
- eng
- Full Text
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