## Duke Mathematical Journal

### Stability and invariant random subgroups

#### Abstract

Consider $\operatorname{Sym}(n)$ endowed with the normalized Hamming metric $d_{n}$. A finitely generated group $\Gamma$ is P-stable if every almost homomorphism $\rho _{n_{k}}\colon\Gamma \rightarrow \operatorname{Sym}(n_{k})$ (i.e., for every $g,h\in \Gamma$, $\lim _{k\rightarrow \infty }d_{n_{k}}(\rho _{n_{k}}(gh),\rho _{n_{k}}(g)\rho _{n_{k}}(h))=0$) is close to an actual homomorphism $\varphi _{n_{k}}\colon \Gamma \rightarrow \operatorname{Sym}(n_{k})$. 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.

#### Article information

Source
Duke Math. J., Volume 168, Number 12 (2019), 2207-2234.

Dates
Revised: 15 January 2019
First available in Project Euclid: 14 August 2019

https://projecteuclid.org/euclid.dmj/1565748248

Digital Object Identifier
doi:10.1215/00127094-2019-0024

Mathematical Reviews number (MathSciNet)
MR3999445

Subjects
Primary: 20B07: General theory for infinite groups
Secondary: 20F69: Asymptotic properties of groups

#### Citation

Becker, Oren; Lubotzky, Alexander; Thom, Andreas. Stability and invariant random subgroups. Duke Math. J. 168 (2019), no. 12, 2207--2234. doi:10.1215/00127094-2019-0024. https://projecteuclid.org/euclid.dmj/1565748248

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