p-localization of Burger-Mozes universal groups

Abstract

The structure theory of locally compact groups can, to a large extent, be reduced to the study of totally disconnected such groups. This talk concerns an attempt to take a further reduction step via p-groups. We recall the concept of prime localization of totally disconnected locally compact groups first introduced by Colin Reid in 2011: For every such group G and prime p, the p-localization of G is a virtually pro-p group which maps continuously and injectively into G with dense image, and which behaves nicely with respect to the scale and modular function. The talk aims to determine said prime localization for Burger-Mozes universal groups acting on regular trees locally like a given permutation group. A short discussion of these groups is followed by the main statement relating the localization to Le Boudec groups acting on the same tree with almost prescribed local action and ideas of proof.