Growing trees from compact subgroups

TL;DR

This paper links local subgroup structures to the global topology of compactly generated totally disconnected locally compact groups, providing conditions for multiple ends and subgroup triviality through actions on trees.

Contribution

It introduces a new condition connecting subgroup properties to the group's end structure, involving actions on trees and Boolean algebra of centralisers.

Findings

Condition for multiple ends based on compact subgroups

Action of a quotient group on a tree with boundary support

Criteria for trivial or open direct factors in one-ended groups

Abstract

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups $G$, finding a sufficient condition for $G$ to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group $G/N$ on a tree with faithful micro-supported action on the boundary, where $N$ is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of $G/N$ with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group $G$, for all direct factors of open subgroups of $G$ to be trivial or open.