Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-27T23:37:16.093Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuum of allosteric actions for non-amenable surface groups

Part of: Structure and classification of infinite or finite groups Topological dynamics Ergodic theory

Published online by Cambridge University Press:  19 July 2023

MATTHIEU JOSEPH Open the ORCID record for MATTHIEU JOSEPH [Opens in a new window]
MATTHIEU JOSEPH*
Affiliation:
Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, Orsay 91405, France
*
e-mail: matthieu.joseph@universite-paris-saclay.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $\Sigma $ be a closed surface other than the sphere, the torus, the projective plane or the Klein bottle. We construct a continuum of probability measure preserving ergodic minimal profinite actions for the fundamental group of $\Sigma $ that are topologically free but not essentially free, a property that we call allostery. Moreover, the invariant random subgroups we obtain are pairwise distincts.

Keywords

measure-preserving actionsminimal actionsinvariant random subgroupsuniformly recurrent subgroupsspace of subgroups

MSC classification

Primary: 37A15: General groups of measure-preserving transformations 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 20E08: Groups acting on trees 20E26: Residual properties and generalizations; residually finite groups 20E15: Chains and lattices of subgroups, subnormal subgroups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 16
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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.)

References

Abért, M. and Elek, G.. Non-abelian free groups admit non-essentially free actions on rooted trees. Preprint, 2007, arXiv:0707.0970.Google Scholar
Abért, M. and Elek, G.. Hyperfinite actions on countable sets and probability measure spaces. Dynamical Systems and Group Actions. Dedicated to Anatoli Stepin on the Occasion of his 70th Birthday. Eds. L. Bowen, R. Grigorchuk and Y. Vorobets. American Mathematical Society, Providence, RI, 2012, pp. 1–16.Google Scholar
Abért, M., Glasner, Y. and Virág, B.. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163(3) (2014), 465–488.CrossRefGoogle Scholar
Baumslag, G.. On generalised free products. Math. Z. 78 (1962), 423–438.CrossRefGoogle Scholar
Baumslag, B.. Residually free groups. Proc. Lond. Math. Soc. (3) 17 (1967), 402–418.CrossRefGoogle Scholar
Bekka, B.. Character rigidity of simple algebraic groups. Math. Ann. 378 (2020), 1223–1243.CrossRefGoogle Scholar
Bergeron, N. and Gaboriau, D.. Asymptotique des nombres de Betti, invariants ${\ell}^2$ et laminations. Comment. Math. Helv. 79(2) (2004), 362–395.CrossRefGoogle Scholar
Bowen, L., Grigorchuk, R. and Kravchenko, R.. Characteristic random subgroups of geometric groups and free abelian groups of infinite rank. Trans. Amer. Math. Soc. 369(2) (2017), 755–781.CrossRefGoogle Scholar
Becker, O., Lubotzky, A. and Thom, A.. Stability and invariant random subgroups. Duke Math. J. 168(12) (2019), 2207–2234.CrossRefGoogle Scholar
Carderi, A., Gaboriau, G. and Le Maître, F.. On dense totipotent free subgroups in full groups. Geom. Topol., accepted for publication, 2020, arXiv:2009.03080.Google Scholar
Creutz, D. and Peterson, J.. Stabilizers of ergodic actions of lattices and commensurators. Trans. Amer. Math. Soc. 369(6) (2017), 4119–4166.CrossRefGoogle Scholar
Creutz, D.. Stabilizers of actions of lattices in products of groups. Ergod. Th. & Dynam. Sys. 37(4) (2017), 1133–1186.CrossRefGoogle Scholar
Dudko, A. and Medynets, K.. Finite factor representations of Higman–Thompson groups. Groups Geom. Dyn. 8(2) (2014), 375–389.CrossRefGoogle Scholar
Grigorchuk, R., Nekrashevich, V. and Sushchanskii, V.. Automata, dynamical systems, and groups. Dynamical Systems, Automata, and Infinite Groups. Ed. R. Grigorchuk. MAIK Nauka/Interperiodica Publishing, Moscow, 2000, pp. 128–203. Translation from the Russian.Google Scholar
Grigorchuk, R.. Some topics in the dynamics of group actions on rooted trees. Proc. Steklov Inst. Math. 273 (2011), 64–175.CrossRefGoogle Scholar
Gruenberg, K. W.. Residual properties of infinite soluble groups. Proc. Lond. Math. Soc. (3) 7 (1957), 29–62.CrossRefGoogle Scholar
Glasner, E. and Weiss, B.. Uniformly recurrent subgroups. Recent Trends in Ergodic Theory and Dynamical Systems. International Conference in Honor of S. G. Dani’s 65th Birthday, Vadodara, India, December 26–29, 2012. Proceedings. Eds. S. Bhattacharya, T. Das, A. Ghosh and R. Shah. American Mathematical Society, Providence, RI, 2015, pp. 63–75.Google Scholar
Kim, G. and McCarron, J.. On amalgamated free products of residually $p$ -finite groups. J. Algebra 162(1) (1993), 1–11.CrossRefGoogle Scholar
Le Boudec, A. and Matte Bon, N.. Subgroup dynamics and ${C}^{\ast }$ -simplicity of groups of homeomorphisms. Ann. Sci. Éc. Norm. Supér. (4) 51(3) (2018), 557–602.CrossRefGoogle Scholar
Peterson, J. and Thom, A.. Character rigidity for special linear groups. J. Reine Angew. Math. 716 (2016), 207–228.CrossRefGoogle Scholar
Serre, J.-P.. Arbres, Amalgames, ${SL}_2$ (Astérisques, 46). Société Mathématique de France (SMF), Paris, 1977.Google Scholar
Stuck, G. and Zimmer, R. J.. Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (2) 139(3) (1994), 723–747.CrossRefGoogle Scholar
Takahasi, M.. Note on chain conditions in free groups. Osaka J. Math. 3 (1951), 221–225.Google Scholar
Thomas, S. and Tucker-Drob, R.. Invariant random subgroups of inductive limits of finite alternating groups. J. Algebra 503 (2018), 474–533.CrossRefGoogle Scholar
Vershik, A. M.. Totally nonfree actions and the infinite symmetric group. Mosc. Math. J. 12(1) (2012), 193–212.CrossRefGoogle Scholar