GdT Actions !

Colin Reid (in presence) : Building trees out of compact subgroups


Abstract: (Joint work with Pierre-Emmanuel Caprace and Timothée Marquis.)  Compactly generated locally compact groups G have a well-behaved notion of ends, generalizing the number of ends of a finitely generated group: G has 0,1,2 or infinitely many ends, and having more than one end is associated with a certain kind of action on a tree (not necessarily of finite degree).  It can also happen that the action on the tree is micro-supported, meaning that for each half-tree, there is an element fixing that half-tree pointwise but acting nontrivially on the opposite half-tree.  The existence of micro-supported actions is in turn closely related to the structure of locally normal subgroups (closed subgroups with open normalizer) and has further implications for global properties of G, for instance it often leads to a nonamenable action of G on the Cantor space.

We find a sufficient condition in terms of a conjugacy class of compact subgroups for G to act on a tree in a way that shows G has infinitely many ends, and the action is also micro-supported.  As an application, we obtain a connection between local and large-scale structure, for a class of groups acting on buildings that are obtained from Kac–Moody groups over finite fields.  This is a class of totally disconnected locally compact groups where much is known about the group on a large scale (via the geometry of the building), but the structure of the compact open subgroups is still mysterious.