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.
Acknowledgements
We are extremely grateful to the anonymous referee, who did an impressive job reading our paper in detail and made a variety of insightful observations, careful remarks and helpful suggestions. In particular, Proposition 3.13, Lemma 4.5 and Remark 5.4 would not have seen the light without the referee’s feedback.
Communicated by: Anna Wienhard
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