Stability and invariant random subgroups

Consider Sym.n/ endowed with the normalized Hamming metric d_n. A finitely generated group Γ is P-stable if every almost homomorphism ρ_nk: Γ → Sym(n_k) (i.e., for every g;h ϵ Γ, lim_k→∞d_nk(ρ_nk(gh),ρ_nk(g)ρ_nk(h))= 0) is close to an actual homomorphism ψ_nk: Γ → Sym(n_k). Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and P?aunescu 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.