Locally normal subgroups of totally disconnected groups. Part I: General theory

TL;DR

This paper develops a general theory of locally normal subgroups in totally disconnected, locally compact groups, introducing structure and centraliser lattices, and analyzing their algebraic and topological properties.

Contribution

It introduces the concept of the structure lattice and centraliser lattice for locally normal subgroups, and explores their properties and implications for the topology of the group.

Findings

The structure lattice forms a modular lattice modulo commensurability.

The canonical maximal quotient H has a Boolean algebra of centralisers.

For second-countable H, the topology is determined by its algebraic structure.

Abstract

Let G be a totally disconnected, locally compact group. A closed subgroup of G is locally normal if its normaliser 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, called the structure lattice of G. We show that G admits a canonical maximal quotient H for which the quasi-centre and the abelian locally normal subgroups are trivial. In this situation the structure lattice of H has a canonical subset called the centraliser lattice, forming a Boolean algebra whose elements correspond to centralisers of locally normal subgroups. If H is second-countable and acts faithfully on its centraliser lattice, we show that the topology of H is determined by its algebraic structure (and thus invariant by every abstract group automomorphism), and also that the action on the Stone space of the centraliser 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.