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Simple locally compact groups acting on trees and their germs of automorphisms

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Abstract

Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours.

We study the interplay between these different aspects for the special class of groups satisfying Titsʼ independence property. We show that such a group has few open subgroups if and only if it acts locally primitively. Moreover, we show that it always admits many germs of automorphisms which do not extend to automorphisms, from which we deduce a negative answer to a question by George Willis. Finally, under suitable assumptions, we compute the full group of germs of automorphisms; in some specific cases, these turn out to be simple and compactly generated, thereby providing a new infinite family of examples which generalise Neretinʼs group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups.

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Authors and Affiliations

  1. UCLouvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium

    Pierre-Emmanuel Caprace*

  2. Ghent University Department of Mathematics, Krijgslaan, 281–S22 9000, Gent, Belgium

    Tom De Medts

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  1. Pierre-Emmanuel Caprace*
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  2. Tom De Medts
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Correspondence to Pierre-Emmanuel Caprace*.

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*F.R.S.-FNRS Research Associate.

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Caprace*, PE., De Medts, T. Simple locally compact groups acting on trees and their germs of automorphisms. Transformation Groups 16, 375–411 (2011). https://doi.org/10.1007/s00031-011-9131-z

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  • DOI: https://doi.org/10.1007/s00031-011-9131-z

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