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

Abstract

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.

Keywords

• 22D05 (primary)
• 20E15 (secondary)