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Restricted universal groups for right-angled buildings

Jens Bossaert and Tom De Medts

Vol. 20 (2023), No. 2-3, 177–208
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 a main result, a precise criterion for when these groups are virtually simple.

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Keywords
right-angled buildings, universal groups, locally compact groups, simple groups
Mathematical Subject Classification
Primary: 22D05, 22F50, 51E24
Milestones
Received: 25 July 2022
Revised: 22 December 2022
Accepted: 24 January 2023
Published: 13 September 2023
Authors
Jens Bossaert
Department of Mathematics: Algebra and Geometry
Ghent University
Ghent
Belgium
Tom De Medts
Department of Mathematics: Algebra and Geometry
Ghent University
Ghent
Belgium