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

Part of:Ergodic theorySpecial aspects of infinite or finite groups

Published online by Cambridge University Press:  30 April 2021

ARIE LEVITOpen the ORCID record for ARIE LEVIT[Opens in a new window] and
ALEXANDER LUBOTZKYOpen the ORCID record for ALEXANDER LUBOTZKY[Opens in a new window]
ARIE LEVIT
Affiliation:
Yale University, Department of Mathematics, New Haven, CT, USAInstitute of Mathematics, Hebrew University, Givat-ram, Jerusalem 91904, Israel
ALEXANDER LUBOTZKY*
Affiliation:
Yale University, Department of Mathematics, New Haven, CT, USAInstitute of Mathematics, Hebrew University, Givat-ram, Jerusalem 91904, Israel
*
e-mail: alex.lubotzky@mail.huji.ac.il
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Abstract

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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 groups20F16: Solvable groups, supersolvable groups37A99: 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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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