Stability and invariant random subgroups

Consider Sym(n) endowed with the normalized Hamming metric d(n). A finitely generated group F is P-stable if every almost homomorphism rho(nk) : Gamma -> Sym(n(k)) (i.e., for every g,h is an element of Gamma, lim(k ->infinity) d(nk) (rho(nk) (gh), rho(nk) (g)rho(nk )(h )) = 0) is close to an actual homomorphism phi(nk) : Gamma -> Sym(n(k)). Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and Paunescu 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.