Growing trees from compact subgroups

Caprace, Pierre-Emmanuel [UCL] Marquis, Timothée [UCL] Reid, Colin D.

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 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 locally normal subgroups of G/N. 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.