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 $\mathcal{S}$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are non-discrete. Given $G\in \mathcal{S}$, 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 $\mathcal{S}$ abstractly simple? Can a group in $\mathcal{S}$ be amenable? Can a group in $\mathcal{S}$ be such that the contraction groups of all of its elements are trivial?

82 pages

References in corpus (23)

• Metric geometry of locally compact groups
• Simplicite de groupes d'automorphismes d'espaces a courbure negative
• Decomposing locally compact groups into simple pieces
• Generators and commutators in finite groups; abstract quotients of compact groups
• Elementary totally disconnected locally compact groups
• Simple locally compact groups acting on trees and their germs of automorphisms
• A product for permutation groups and topological groups
• Locally normal subgroups of totally disconnected groups. Part I: General theory
• Abstract commensurators of profinite groups
• The Nub of an Automorphism of a Totally Disconnected, Locally Compact Group
• Finitely presented groups associated with expanding maps
• Algebraic properties of profinite groups
• Automorphism groups of right-angled buildings: simplicity and local splittings
• On the congruence subgroup problem for branch groups
• Trees, contraction groups, and Moufang sets
• Totally disconnected locally compact groups with a linear open subgroup
• Abstract simplicity of locally compact Kac-Moody groups
• Locally normal subgroups of simple locally compact groups
• Finiteness properties of profinite groups
• Contraction groups in complete Kac-Moody groups
• The profinite completion of a group localised at a subgroup
• On regular CAT(0) cube complexes
• Topological rigidity in totally disconnected locally compact groups

Cited by in corpus (18)

• Metric geometry of locally compact groups
• A product for permutation groups and topological groups
• Locally normal subgroups of totally disconnected groups. Part I: General theory
• New simple lattices in products of trees and their projections
• Totally disconnected locally compact groups with a linear open subgroup
• A new lattice invariant for lattices in totally disconnected locally compact groups
• Bounding the covolume of lattices in products
• Locally compact groups whose ergodic or minimal actions are all free
• Around the Lie correspondence for complete Kac-Moody groups and Gabber-Kac simplicity
• Dynamics of flat actions on totally disconnected, locally compact groups
• Commensurated subgroups and micro-supported actions
• Neretin groups admit no non-trivial invariant random subgroups
• Discrete locally finite full groups of Cantor set homeomorphisms
• Future directions in locally compact groups
• Growing trees from compact subgroups
• A class of well-founded totally disconnected locally compact groups
• Topologically simple, totally disconnected, locally compact infinite matrix groups
• A sufficient condition for a locally compact almost simple group to have open monolith