Simple locally compact groups acting on trees and their germs of automorphisms
By P. Caprace, Tom De Medts. 2010
52 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b353f4909c50f3b0d22fa51e1ff4ef0cc81549eb
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
- Adrien Le Boudec
- Colin D. Reid
- George A. Willis
- P. Caprace
- Denis V. Osin
- Daan Krammer
- Anthony Genevois
- K. Fujiwara
Citations in the corpus
- LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART II: COMPACTLY GENERATED SIMPLE GROUPS (2014, methodology): “Some variations have been constructed by Barnea– Ershov–Weigel [5] and Caprace–De Medts [15].”
- Groups acting on trees with almost prescribed local action (2015, background): “In [13] the authors proved that whenF is 2-transitive and every point stabilizerFa in F is equal to its normalizer in Sym.”