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.
Highlights
The aim of the present article is to establish foundations for a study of general nondiscrete totally disconnected locally compact (t.d.l.c.) groups in terms of their local structure
Inspired by earlier work of Wilson [26] on just-infinite groups and by Barnea et al [2] on abstract commensurators of profinite groups, we develop properties of this lattice
While a rather rich and deep theory was developed for some special classes of t.d.l.c. groups, for a long time the only general result known was van Dantzig’s theorem, which asserts that compact open subgroups exist and form a basis of identity neighbourhoods
Summary
The aim of the present article is to establish foundations for a study of general nondiscrete totally disconnected locally compact (t.d.l.c.) groups in terms of their local structure. Caprace and Monod [7] identified simple pieces of a t.d.l.c. group G under the hypothesis that G has no discrete quotients while, at the local level, Burger and Mozes [6] introduced, in analogy with the kernel of the adjoint representation of a Lie group, the quasicentre of G, which contains in particular any discrete normal subgroup of G The former of the two motivates the focus of our paper [8] on simple groups, while the latter was important in the approach introduced in [2] and is developed further in Sections 3 and 6 below as part of the foundation for a general theory of the local structure of t.d.l.c. groups. Its structure lattice is a local invariant of a t.d.l.c. group G, since it is determined by any open compact subgroup, which admits a natural action of the topological automorphism group of G. The topology of G is the unique σ -compact locally compact group topology on G, and the set of all locally compact group topologies of G is in natural bijection with the set of fixed points of G acting on LN (G) by conjugation
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