Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups

Abstract

We use the structure lattice, introduced in Part I, to undertake a systematic study of the class $\mathscr S$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are non-discrete. Given $G \in \mathscr S$, we show that compact open subgroups of $G$ involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centraliser lattice and local decomposition lattice of $G$ are Boolean algebras. We show that the $G$-action on the Stone space of those Boolean algebras is minimal, strongly proximal, and micro-supported. Building upon those results, we obtain partial answers to the following key problems: Are all groups in $\mathscr S$ abstractly simple? Can a group in $\mathscr S$ be amenable? Can a group in $\mathscr S$ be such that the contraction groups of all of its elements are trivial?