Universal groups for right-angled buildings

TL;DR

This paper extends the concept of universal groups from trees to right-angled buildings, demonstrating simplicity and describing their structure as locally compact groups under certain conditions.

Contribution

It generalizes universal groups to right-angled buildings and establishes their simplicity and structural properties in the locally finite case.

Findings

Universal groups are simple under specified conditions.

Universal groups are compactly generated totally disconnected locally compact groups.

Maximal compact open subgroups are described as limits of generalized wreath products.

Abstract

In 2000, M. Burger and S. Mozes introduced universal groups acting on trees with a prescribed local action. We generalize this concept to groups acting on right-angled buildings. When the right-angled building is thick and irreducible of rank at least 2 and each of the local permutation groups is transitive and generated by its point stabilizers, we show that the corresponding universal group is a simple group. When the building is locally finite, these universal groups are compactly generated totally disconnected locally compact groups, and we describe the structure of the maximal compact open subgroups of the universal groups as a limit of generalized wreath products.