Open subgroups of the automorphism group of a right-angled building

Abstract

We study the group of type-preserving automorphisms of a right-angled building, in particular when the building is locally finite. Our aim is to characterize the proper open subgroups as the finite index closed subgroups of the stabilizers of proper residues. One of the main tools is the new notion of firm elements in a right-angled Coxeter group, which are those elements for which the final letter in each reduced representation is the same. We also introduce the related notions of firmness for arbitrary elements of such a Coxeter group and n-flexibility of chambers in a right-angled building. These notions and their properties are used to determine the set of chambers fixed by the fixator of a ball. Our main result is obtained by combining these facts with ideas by Pierre-Emmanuel Caprace and Timothée Marquis in the context of Kac–Moody groups over finite fields, where we had to replace the notion of root groups by a new notion of root wing groups.

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Acknowledgements

This paper would never have existed without the help of Pierre-Emmanuel Caprace. Not only did he suggest the study of open subgroups of the automorphism group of right-angled buildings to us; we also benefited a lot from discussions with him. Two anonymous referees did a great job in pointing out inaccuracies and suggesting improvements for the exposition. We also thank the Research Foundation in Flanders (F.W.O.-Vlaanderen) for their support through the project “Automorphism groups of locally finite trees” (G011012N).

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Correspondence to Tom De Medts.

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De Medts, T., Silva, A.C. Open subgroups of the automorphism group of a right-angled building. Geom Dedicata 203, 1–23 (2019). https://doi.org/10.1007/s10711-019-00423-7

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Keywords

  • Right-angled buildings
  • Right-angled Coxeter groups
  • Totally disconnected locally compact groups
  • open subgroups

Mathematics Subject Classification

  • 51E24
  • 22D05
  • 20F65