Consider endowed with the normalized Hamming metric . A finitely generated group is P-stable if every almost homomorphism (i.e., for every , ) is close to an actual homomorphism . 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.
"Stability and invariant random subgroups." Duke Math. J. 168 (12) 2207 - 2234, 1 September 2019. https://doi.org/10.1215/00127094-2019-0024