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CHABAUTY LIMITS OF SIMPLE GROUPS ACTING ON TREES

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Locally compact groups and their algebras

Published online by Cambridge University Press:  06 August 2018

Pierre-Emmanuel Caprace  and
Nicolas Radu
Pierre-Emmanuel Caprace
Affiliation:
UCLouvain, 1348 Louvain-la-Neuve, Belgium ([email protected])
Nicolas Radu
Affiliation:
UCLouvain, 1348 Louvain-la-Neuve, Belgium ([email protected])
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Abstract

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\text{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree ${\geqslant}3$, then the set of isomorphism classes of topologically simple closed subgroups of $\text{Aut}(T)$ acting doubly transitively on $\unicode[STIX]{x2202}T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

Keywords

simple grouplocally compact groupChabauty spacegroups acting on trees

MSC classification

Primary: 20E08: Groups acting on trees 20E32: Simple groups 20E42: Groups with a $BN$-pair; buildings 20F65: Geometric group theory 22D05: General properties and structure of locally compact groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1093 - 1120
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Copyright
© Cambridge University Press 2018

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Footnotes

†

F.R.S.-FNRS Senior Research Associate, supported in part by the ERC (grant no. 278469).

‡

F.R.S.-FNRS Research Fellow.

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