Restricted universal groups for right-angled buildings

TL;DR

This paper explores restricted universal groups acting on right-angled buildings, combining ideas from groups on trees and geometric structures, and provides criteria for their simplicity.

Contribution

It introduces the study of restricted universal groups on right-angled buildings and establishes conditions for their simplicity.

Findings

Groups exhibit specific permutational properties

Topological properties are characterized

Simplicity criteria are precisely determined

Abstract

In 2000, Marc Burger and Shahar Mozes introduced universal groups acting on trees. Such groups provide interesting examples of totally disconnected locally compact groups. Intuitively, these are the largest groups for which all local actions satisfy a prescribed behavior. Since then, their study has evolved in various directions. In particular, Adrien Le Boudec has studied restricted universal groups, where the prescribed behavior is allowed to be violated in a finite number of vertices. On the other hand, we have been studying universal groups acting on right-angled buildings, a class of geometric objects with a much more general structure than trees. The aim of the current paper is to combine both ideas: we will study restricted universal groups acting on right-angled buildings. We show several permutational and topological properties of those groups, with as main result a precise criterion for when these groups are simple.