Growing trees from compact subgroups

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.

24 pages; journal accepted version. To appear in Groups, Geometry and Dynamics

References in corpus (5)

• Metric geometry of locally compact groups
• Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups
• A product for permutation groups and topological groups
• Groups acting on trees with Tits' independence property (P)
• Scale Groups