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RADU GROUPS ACTING ON TREES ARE CCR

Part of: Low-dimensional topology Locally compact groups and their algebras Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  06 March 2024

LANCELOT SEMAL
LANCELOT SEMAL*
Affiliation:
UCLouvain, 1348 Louvain-la-Neuve, Belgium
*
e-mail: lancelot.semal@uclouvain.be
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Abstract

We classify the irreducible unitary representations of closed simple groups of automorphisms of trees acting $2$-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, we confirm Claudio Nebbia’s CCR conjecture on trees for $(d_0,d_1)$-semi-regular trees such that $d_0,d_1\in \Theta $, where $\Theta $ is an asymptotically dense set of positive integers.

Keywords

unitary representationsgroups of automorphisms of semi-regular treestype I groupsNebbia’s conjecture

MSC classification

Primary: 22D12: Other representations of locally compact groups
Secondary: 20E08: Groups acting on trees 57M07: Topological methods in group theory

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 117 , Issue 2 , October 2024 , pp. 149 - 186
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by George Willis

The author is an F.R.S.-FNRS Research Fellow.

References

Amann, O. E., ‘Groups of tree-automorphisms and their unitary representations’, PhD Thesis, ETH Zurich, 2003.Google Scholar
Bekka, B. and de la Harpe, P., Unitary Representations of Groups, Duals, and Characters, Mathematical Surveys and Monographs, 250 (American Mathematical Society, Providence, RI, 2020).CrossRefGoogle Scholar
Bernšteĭn, I. N., ‘All reductive $p$ -adic groups are of type I’, Funktsional. Anal. i Prilozhen. 8(2) (1974), 3–6.Google Scholar
Banks, C., Elder, M. and Willis, G. A., ‘Simple groups of automorphisms of trees determined by their actions on finite subtrees’, J. Group Theory 18(2) (2015), 235–261.CrossRefGoogle Scholar
Bouaziz-Kellil, F., ‘Représentations sphériques des groupes agissant transitivement sur un arbre semi-homogène’, Bull. Soc. Math. France 116(3) (1988), 255–278.CrossRefGoogle Scholar
Burger, M. and Mozes, S., ‘Groups acting on trees: from local to global structure’, Publ. Math. Inst. Hautes Études Sci. 92 (2000), 113–150.CrossRefGoogle Scholar
Caprace, P.-E. and Ciobotaru, C., ‘Gelfand pairs and strong transitivity for Euclidean buildings’, Ergodic Theory Dynam. Systems 35(4) (2015), 1056–1078.CrossRefGoogle Scholar
Choucroun, F. M., ‘Analyse harmonique des groupes d’automorphismes d’arbres de Bruhat–Tits’, Mém. Soc. Math. Fr. (N.S.) 58 (1994), 170.Google Scholar
Ciobotaru, C., ‘A note on type I groups acting on d-regular trees’, Preprint, 2015, arXiv:1506.02950.Google Scholar
Caprace, P.-E., Kalantar, M. and Monod, N., ‘A type I conjecture and boundary representations of hyperbolic groups’, Proc. Lond. Math. Soc. 127(2) (2023), 447–486.CrossRefGoogle Scholar
Cartwright, D. I., Młotkowski, W. and Steger, T., ‘Property (T) and $\tilde{A_2}$ groups’, Ann. Inst. Fourier (Grenoble) 44(1) (1994), 213–248.CrossRefGoogle Scholar
Figà-Talamanca, A. and Nebbia, C., Harmonic Analysis and Representation Theory for Groups Acting on Homogeneous Trees, London Mathematical Society Lecture Note Series, 162 (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
Harish-Chandra, Harmonic Analysis on Reductive $p$ -adic Groups, Lecture Notes in Mathematics, 162 (Springer-Verlag, Berlin–New York, 1970); notes by G. van Dijk.Google Scholar
Houdayer, C. and Raum, S., ‘Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras’, Comment. Math. Helv. 94(1) (2019), 185–219.CrossRefGoogle Scholar
Iozzi, A. and Picardello, M. A., ‘Spherical functions on symmetric graphs’, in Harmonic Analysis (Cortona, 1982), Lecture Notes in Mathematics, 992 (eds. Mauceri, G., Ricci, F. and Weiss, G.) (Springer, Berlin, 1983), 344–386.CrossRefGoogle Scholar
Kallman, R. R., ‘Certain topological groups are type I’, Bull. Amer. Math. Soc. (N.S.) 76 (1970), 404–406.CrossRefGoogle Scholar
Kaniuth, E. and Taylor, K. F., Induced Representations of Locally Compact Groups, Cambridge Tracts in Mathematics, 197 (Cambridge University Press, Cambridge, 2013).Google Scholar
Lang, S., SL2(R), Graduate Texts in Mathematics, 197 (Springer-Verlag, New York, 1985); reprint of the 1975 edition.CrossRefGoogle Scholar
Mackey, G. W., The Theory of Unitary Group Representations (University of Chicago Press, Chicago–London, 1976); based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, IL, 1955, Chicago Lectures in Mathematics.Google Scholar
Matsumoto, H., ‘Fonctions sphériques sur un groupe semi-simple p-adique’, C. R. Math. Acad. Sci. Paris 269 (1969), 829–832.Google Scholar
Matsumoto, H., Analyse Harmonique dans les Systèmes de Tits Bornologiques de Type Affine, Lecture Notes in Mathematics, 590 (Springer-Verlag, Berlin–New York, 1977).CrossRefGoogle Scholar
Nebbia, C., ‘Groups of isometries of a tree and the CCR property’, Rocky Mountain J. Math. 29(1) (1999), 311–316.CrossRefGoogle Scholar
Ol’shanskii, G. I., ‘Classification of the irreducible representations of the automorphism groups of Bruhat–Tits trees’, Funktsional. Anal. i Prilozhen. 11(1) (1977), 32–42, 96.Google Scholar
Ol’shanskii, G. I., ‘New “large” groups of type I’, J. Math. Sci. 18 (1982), 22–39. (Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1980), 31–52, 228 (in Russian).Google Scholar
Radu, N., ‘A classification theorem for boundary 2-transitive automorphism groups of trees’, Invent. Math. 209(1) (2017), 1–60.CrossRefGoogle Scholar
Semal, L., ‘Unitary representations of totally disconnected locally compact groups satisfying Ol’shanskii’s factorisation’, Repr. Theory 27(11) (2023), 356–414.CrossRefGoogle Scholar
Smith, S. M., ‘A product for permutation groups and topological groups’, Duke Math. J. 166(15) (2017), 2965–2999.CrossRefGoogle Scholar