Stability and Invariant Random Subgroups

Abstract

Consider $\mathrm{Sym}(n)$, endowed with the normalized Hamming metric $d_n$. A finitely-generated group $\Gamma$ is \emph{P-stable} if every almost homomorphism ${\rho }_{n_k}:\Gamma \to \mathrm{Sym}(n_k)$ (i.e., for every $g,h\in \Gamma$, ${\lim }_{k\to \infty }{d}_{n_k}({\rho }_{n_k}(gh),{\rho }_{n_k}(g){\rho }_{n_k}(h))=0$) is close to an actual homomorphism ${\phi }_{n_k}:\Gamma \to \mathrm{Sym}(n_k)$. 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 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.

24 pages; v2 includes minor updates and new references