Hostname: page-component-cb9f654ff-qc88w Total loading time: 0 Render date: 2025-09-04T08:41:56.023Z Has data issue: false hasContentIssue false
  • English
  • Français

DECOMPOSITION THEOREMS FOR AUTOMORPHISM GROUPS OF TREES

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

Published online by Cambridge University Press:  20 May 2020

MAX CARTER Open the ORCID record for MAX CARTER [Opens in a new window]  and
GEORGE A. WILLIS Open the ORCID record for GEORGE A. WILLIS [Opens in a new window]
MAX CARTER
Affiliation:
School of Mathematical and Physical Sciences,University of Newcastle, Callaghan, New South Wales, Australia email max.carter@newcastle.edu.au
GEORGE A. WILLIS*
Affiliation:
School of Mathematical and Physical Sciences,University of Newcastle, Callaghan, New South Wales, Australia email george.willis@newcastle.edu.au
*
✉george.willis@newcastle.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

Keywords

groups acting on treesautomorphism groupregular treelabel-regular treeLie group decomposition

MSC classification

Primary: 20E08: Groups acting on trees
Secondary: 22D05: General properties and structure of locally compact groups

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 104 - 112
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

This research was supported by the Australian Research Council grant FL170100032.

References

Baumgartner, U. and Willis, G., ‘Contraction groups and scales of automorphisms of totally disconnected locally compact groups’, Israel J. Math. 142 (2004), 221–248.10.1007/BF02771534CrossRefGoogle Scholar
Benoist, Y. and Oh, H., ‘Polar decomposition for p-adic symmetric spaces’, Int. Math. Res. Not. 2007(24) (2007), Article ID rnm121.Google Scholar
Bruhat, F., ‘Sur une classe de sous-groupes de Chevalley sur un corps p-adique’, Publ. Math. Inst. Hautes Études Sci. 23 (1964), 45–74.10.1007/BF02684310CrossRefGoogle Scholar
Caprace, P.-E., Reid, C. D. and Willis, G. A., ‘Locally normal subgroups of totally disconnected groups. Parts I and II’, Forum Math. Sigma 5 (2017), e11, 76 pages, and e12, 89 pages.Google Scholar
Delorme, P. and Sécherre, V., ‘An analogue of the Cartan decomposition for p-adic symmetric spaces of split p-adic reductive groups’, Pacific J. Math. 251 (2011), 1–21.10.2140/pjm.2011.251.1CrossRefGoogle Scholar
Glöckner, H., ‘Real and p-adic Lie algebra functors on the category of topological groups’, Pacific J. Math. 203 (2002), 321–368.10.2140/pjm.2002.203.321CrossRefGoogle Scholar
Glöckner, H., ‘Locally compact groups built up from p-adic Lie groups, for p in a given set of primes’, J. Group Theory 9 (2006), 427–454.10.1515/JGT.2006.028CrossRefGoogle Scholar
Iwahori, N. and Matsumoto, H., ‘On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups’, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 5–48.10.1007/BF02684396CrossRefGoogle Scholar
Knapp, A. W., Lie Groups Beyond an Introduction, 2nd edn (Birkhäuser, Boston–Basel–Berlin, 2002).Google Scholar
Le Boudec, A., ‘Groups acting on trees with almost prescribed local action’, Comment. Math. Helv. 91 (2016), 253–293.10.4171/CMH/385CrossRefGoogle Scholar
Le Boudec, A., ‘Groups of automorphisms and almost automorphisms of trees: subgroups and dynamics’, in: 2016 MATRIX Annals, MATRIX Book Series, Vol. 1 (eds. Wood, D., de Gier, J., Praeger, C. and Tao, T.) (Springer, Cham, 2018), 501–523.CrossRefGoogle Scholar
Serre, J.-P., Trees, corrected 1st edn (Springer, Berlin–Heidelberg–New York, 2003).Google Scholar
Tits, J., ‘Sur le groupe des automorphismes d’un arbre’, in: Essays on Topology and Related Topics (Springer, Heidelberg, 1970), 188–211.10.1007/978-3-642-49197-9_18CrossRefGoogle Scholar