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1 September 2019 Stability and invariant random subgroups
Oren Becker, Alexander Lubotzky, Andreas Thom
Duke Math. J. 168(12): 2207-2234 (1 September 2019). DOI: 10.1215/00127094-2019-0024

Abstract

Consider Sym(n) endowed with the normalized Hamming metric dn. A finitely generated group Γ is P-stable if every almost homomorphism ρnk:ΓSym(nk) (i.e., for every g,hΓ, lim kdnk(ρnk(gh),ρnk(g)ρnk(h))=0) is close to an actual homomorphism φnk:ΓSym(nk). Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and Păunescu showed the same for abelian groups and raised many questions, especially about the P-stability of amenable groups. We develop P-stability in general and, in particular, for amenable groups. Our main tool is the theory of invariant random subgroups, which enables us to give a characterization of P-stability among amenable groups and to deduce the stability and instability of various families of amenable groups.

Citation

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Oren Becker. Alexander Lubotzky. Andreas Thom. "Stability and invariant random subgroups." Duke Math. J. 168 (12) 2207 - 2234, 1 September 2019. https://doi.org/10.1215/00127094-2019-0024

Information

Received: 5 March 2018; Revised: 15 January 2019; Published: 1 September 2019
First available in Project Euclid: 14 August 2019

zbMATH: 07145001
MathSciNet: MR3999445
Digital Object Identifier: 10.1215/00127094-2019-0024

Subjects:
Primary: 20B07
Secondary: 20F69

Keywords: group stability , invariant random subgroup

Rights: Copyright © 2019 Duke University Press

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Vol.168 • No. 12 • 1 September 2019
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