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 nondiscrete. 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 centralizer 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 microsupported. 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?
Highlights
The solution to Hilbert’s fifth problem [43, Theorem 4.6] ensures that a connected locally compact group is an inverse limit of Lie groups
For any compactly generated t.d.l.c. group G, we observe (Proposition 4.6) that each compact open subgroup of G/K has finitely many isomorphism types of composition factors, where K is a compact normal subgroup that can be taken to lie in any given identity neighbourhood
Suppose that at least one of the following holds: (i) LD(G) = {0, ∞}; (ii) There is a compact open subgroup U of G such that U is finitely generated as a profinite group and [U, U ] is open in G; (iii) Some nontrivial compact locally normal subgroup of G is finitely generated as a profinite group and every infinite commensurated compact subgroup of G is open
Summary
For any compactly generated t.d.l.c. group G, we observe (Proposition 4.6) that each compact open subgroup of G/K has finitely many isomorphism types of composition factors, where K is a compact normal subgroup that can be taken to lie in any given identity neighbourhood This observation confirms a conjecture formulated in [69, Section 4] and naturally leads to the notion of the local prime content of G/K , which is the unique finite set η = η(G/K ) of primes such that every compact open subgroup of G/K is virtually pro-η and has an infinite pro- p subgroup for each p ∈ η. In relation with the question of abstract simplicity, we observe that the combination of Proposition I and Theorem J directly implies that a group in S with a nontrivial centralizer lattice has a smallest dense normal subgroup which is abstractly simple. In particular G is not abstractly linear over any field
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