Topological and algebraic properties of universal groups for right-angled buildings

Abstract

Abstract We study universal groups for right-angled buildings. Inspired by Simon Smith’s work on universal groups for trees, we explicitly allow local groups that are not necessarily finite nor transitive. We discuss various topological and algebraic properties in this extended setting. In particular, we characterise when these groups are locally compact, when they are abstractly simple, when they act primitively on residues of the building, and we discuss some necessary and sufficient conditions for the groups to be compactly generated. We point out that there are unexpected aspects related to the geometry and the diagram of these buildings that influence the topological and algebraic properties of the corresponding universal groups.