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Simple groups and irreducible lattices in wreath products

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  07 February 2020

ADRIEN LE BOUDEC
ADRIEN LE BOUDEC*
Affiliation:
UCLouvain, IRMP, Chemin du Cyclotron 2, 1348Louvain-la-Neuve, Belgium CNRS, Unité de Mathématiques Pures et Appliquées, ENS-Lyon, France email adrien.le-boudec@ens-lyon.fr
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Abstract

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$, where $C$ is a finite group and $F$ a non-abelian free group.

Keywords

groups acting on treesirreducible lattices in locally compact groupswreath products

MSC classification

Primary: 20E08: Groups acting on trees 20F65: Geometric group theory 20E32: Simple groups 20E22: Extensions, wreath products, and other compositions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1502 - 1513
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Bartholdi, L., Neuhauser, M. and Woess, W.. Horocyclic products of trees. J. Eur. Math. Soc. (JEMS) 10(3) (2008), 771–816.CrossRefGoogle Scholar
Burger, M. and Mozes, S.. Lattices in product of trees. Publ. Math. Inst. Hautes Études Sci. 92 (2000), 151–194.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 Monod, N.. Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol. 2(4) (2009), 701–746.Google Scholar
Caprace, P.-E. and Monod, N.. A lattice in more than two Kac–Moody groups is arithmetic. Israel J. Math. 190 (2012), 413–444.Google Scholar
Cornulier, Y., Fisher, D. and Kashyap, N.. Cross-wired lamplighter groups. New York J. Math. 18 (2012), 667–677.Google Scholar
Cornulier, Y.. Locally compact wreath products. J. Aust. Math. Soc. 107(1) (2019), 26–52; MR 3978031.CrossRefGoogle Scholar
Dyubina, A.. Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups. Int. Math. Res. Not. IMRN 2000(21) (2000), 1097–1101.Google Scholar
Erschler, A.. Generalized wreath products. Int. Math. Res. Not. IMRN 2006 (2006), Art. ID 57835.Google Scholar
Eskin, A., Fisher, D. and Whyte, K.. Quasi-isometries and rigidity of solvable groups. Pure Appl. Math. Q. 3(4) (2007), 927–947.CrossRefGoogle Scholar
Eskin, A., Fisher, D. and Whyte, K.. Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs. Ann. of Math. (2) 176(1) (2012), 221–260.CrossRefGoogle Scholar
Eskin, A., Fisher, D. and Whyte, K.. Coarse differentiation of quasi-isometries II: Rigidity for Sol and lamplighter groups. Ann. of Math. (2) 177(3) (2013), 869–910.CrossRefGoogle Scholar
Frisch, J., Hartman, Y., Tamuz, O. and Vahidi Ferdowsi, P.. Choquet–Deny groups and the infinite conjugacy class property. Ann. of Math. (2) 190(1) (2019), 307–320.CrossRefGoogle Scholar
Frisch, J., Tamuz, O. and Vahidi Ferdowsi, P.. Strong amenability and the infinite conjugacy class property. Invent. Math. 218(3) (2019), 833–851.CrossRefGoogle Scholar
Glasner, S.. Proximal Flows (Lecture Notes in Mathematics, 517) . Springer, Berlin, 1976.Google Scholar
Le Boudec, A.. Groups acting on trees with almost prescribed local action. Comment. Math. Helv. 91(2) (2016), 253–293.CrossRefGoogle Scholar
Le Boudec, A.. Amenable uniformly recurrent subgroups and lattice embeddings. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2020.2.Google Scholar
Raghunathan, M. S.. Discrete Subgroups of Lie Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 68) . Springer, Berlin, 1972.CrossRefGoogle Scholar