Growing trees from compact subgroups
Pierre-Emmanuel Caprace, Timoth\'ee Marquis, Colin D. Reid

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 , finding a sufficient condition for to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group on a tree with faithful micro-supported action on the boundary, where is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group , for all direct factors of open subgroups of to be trivial or open.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Operator Algebra Research · Advanced Topology and Set Theory
