LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART I: GENERAL THEORY
Let $G$ be a totally disconnected, locally compact group. A closed subgroup of $G$ is locally normal if...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2017-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509417000093/type/journal_article |
| Summary: | 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. |
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| ISSN: | 2050-5094 |