Hostname: page-component-7dd5485656-dk7s8 Total loading time: 0 Render date: 2025-10-28T23:03:50.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

Part of: Special aspects of infinite or finite groups Ergodic theory

Published online by Cambridge University Press:  30 April 2021

ARIE LEVIT Open the ORCID record for ARIE LEVIT [Opens in a new window]  and
ALEXANDER LUBOTZKY Open the ORCID record for ALEXANDER LUBOTZKY [Opens in a new window]
ARIE LEVIT
Affiliation:
Yale University, Department of Mathematics, New Haven, CT, USA Institute of Mathematics, Hebrew University, Givat-ram, Jerusalem 91904, Israel
ALEXANDER LUBOTZKY*
Affiliation:
Yale University, Department of Mathematics, New Haven, CT, USA Institute of Mathematics, Hebrew University, Givat-ram, Jerusalem 91904, Israel
*
e-mail: alex.lubotzky@mail.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

Keywords

stabilityinvariant random groupswreath productsmetabelian groupsamenable groups

MSC classification

Primary: 20F69: Asymptotic properties of groups 20F16: Solvable groups, supersolvable groups 37A99: None of the above, but in this section

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 2028 - 2063
Check for updates
Copyright
© The Author(s), 2021. 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.)

Article purchase

Temporarily unavailable

References

Abért, M., Glasner, Y. and Virág, B.. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163(3) (2014), 465–488.CrossRefGoogle Scholar
Abért, M., Jaikin-Zapirain, A. and Nikolov, N.. The rank gradient from a combinatorial viewpoint. Groups Geom. Dyn. (2011), 213–230.CrossRefGoogle Scholar
Aldous, D. and Lyons, R.. Processes on unimodular random networks. Electron. J. Probab. 12 (2007), 1454–1508.CrossRefGoogle Scholar
Alperin, R. C.. Metabelian wreath products are LERF. Preprint, 2006, arXiv:math/0609611.Google Scholar
Arzhantseva, G. and Păunescu, L.. Almost commuting permutations are near commuting permutations. J. Funct. Anal. 269(3) (2015), 745–757.CrossRefGoogle Scholar
Bowen, L. and Burton, P.. Flexible stability and nonsoficity. Preprint, 2019, arXiv:1906.02172.Google Scholar
Bowen, L., Grigorchuk, R. and Kravchenko, R.. Invariant random subgroups of lamplighter groups. Israel J. Math. 207(2) (2015), 763–782.CrossRefGoogle Scholar
Billingsley, P.. Convergence of Probability Measures. John Wiley & Sons, New York, 2013.Google Scholar
Becker, O., Lubotzky, A. and Thom, A.. Stability and invariant random subgroups. Duke Math. J. 168(12) (2019), 2207–2234.CrossRefGoogle Scholar
Bowen, L. and Nevo, A.. von-Neumann and Birkhoff ergodic theorems for negatively curved groups. Preprint, 2013, arXiv:1303.4109.Google Scholar
De Cornulier, Y.. Finitely presented wreath products and double coset decompositions. Geom. Dedicata 122(1) (2006), 89–108.CrossRefGoogle Scholar
De Chiffre, M., Glebsky, L., Lubotzky, A. and Thom, A.. Stability, cohomology vanishing, and non-approximable groups. Preprint, 2017, arXiv:1711.10238.Google Scholar
Gelander, T.. A lecture on invariant random subgroups. Preprint, 2015, arXiv:1503.08402.Google Scholar
Grigorchuk, R. and Kravchenko, R.. On the lattice of subgroups of the lamplighter group. Internat. J. Algebra Comput. 24(06) (2014), 837–877.CrossRefGoogle Scholar
Gorodnik, A. and Nevo, A.. The Ergodic Theory of Lattice Subgroups (Annals of Mathematics Studies, 190). Princeton University Press, Princeton, NJ, 2009.CrossRefGoogle Scholar
Glebsky, L. and Rivera, L. M.. Almost solutions of equations in permutations. Taiwanese J. Math. 13(2A) (2009), 493–500.CrossRefGoogle Scholar
Gromov, M.. Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1(2) (1999), 109–197.CrossRefGoogle Scholar
Grünberg, K. W. Residual properties of infinite soluble groups. Proc. Lond. Math. Soc. 3(1) (1957), 29–62.CrossRefGoogle Scholar
Hall, P.. A contribution to the theory of groups of prime-power order. Proc. Lond. Math. Soc. 2(1) (1934), 29–95.CrossRefGoogle Scholar
Hall, P.. On the finiteness of certain soluble groups. Proc. Lond. Math. Soc. 3(4) (1959), 595–622.CrossRefGoogle Scholar
Hartman, Y. and Tamuz, O.. Furstenberg entropy realizations for virtually free groups and lamplighter groups. J. Anal. Math. 126(1) (2015), 227–257.CrossRefGoogle Scholar
Lindenstrauss, E.. Pointwise theorems for amenable groups. Invent. Math. 146(2) (2001), 259–295.CrossRefGoogle Scholar
Levit, A. and Lubotzky, A.. Uncountably many permutation stable groups. Preprint, 2019, arXiv:1910.11722.Google Scholar
Lennox, J. C. and Robinson, D. J. S.. The Theory of Infinite Soluble Groups. Clarendon Press, Oxford, 2004.CrossRefGoogle Scholar
Phelps, R. R.. Lectures on Choquet’s Theorem. Springer, Berlin, 2001.CrossRefGoogle Scholar
Stuck, G. and Zimmer, R. J.. Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (1994), 723–747.CrossRefGoogle Scholar
Thom, A.. Finitary approximations of groups and their applications. In Proceedings ICM 2018, Rio de Janeiro. Vol. 3. World Scientific, Singapore, 2019.Google Scholar
Weiss, B.. Sofic groups and dynamical systems. Sankhyā Ser. A (2000), 350–359.Google Scholar
Weiss, B.. Monotileable amenable groups. Trans. Amer. Math. Soc. Ser. 2 202 (2001), 257–262.Google Scholar
Zheng, T.. On rigid stabilizers and invariant random subgroups of groups of homeomorphisms. Preprint, 2019, arXiv:1901.04428.Google Scholar