Stability and Invariant Random Subgroups

TL;DR

This paper explores the concept of P-stability in finitely-generated groups, especially amenable groups, using invariant random subgroups to characterize stability and identify stable and unstable group families.

Contribution

It develops a general theory of P-stability for amenable groups and introduces IRS as a key tool for characterization and analysis.

Findings

Finite groups are P-stable.

Abelian groups are P-stable.

Certain amenable groups are stable or unstable.

Abstract

Consider $\operatorname{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{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 P\u{a}unescu showed the same for abelian groups and raised many questions, especially about 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 (IRS), which enables us to give a characterization of P-stability among amenable groups, and to deduce stability and instability of various families of amenable groups.