Stability and invariant random subgroups

Abstract

Consider $Sym\left(n\right)$ endowed with the normalized Hamming metric $d_n$. A finitely generated group $\Gamma$ is P-stable if every almost homomorphism ${\rho }_{n_k}:\Gamma \to Sym\left(n_k\right)$ (i.e., for every $g,h\in \Gamma$, ${lim\hspace{0.17em}}_{k\to \infty }{d}_{n_k}\left({\rho }_{n_k}\right(gh),{\rho }_{n_k}(g\left){\rho }_{n_k}\right(h\left)\right)=0$) is close to an actual homomorphism ${\phi }_{n_k}:\Gamma \to Sym\left(n_k\right)$. 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.