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CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  06 August 2018

Thierry Stulemeijer
Thierry Stulemeijer*
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany ([email protected])
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Abstract

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

Keywords

group theory and generalizationstopological groupslinear algebraic groups

MSC classification

Primary: 20E08: Groups acting on trees 20E42: Groups with a $BN$-pair; buildings 20F65: Geometric group theory 20G25: Linear algebraic groups over local fields and their integers
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 4 , July 2020 , pp. 1031 - 1091
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Copyright
© Cambridge University Press 2018

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Footnotes

Postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn.

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