Stability and Testability

A countable group $G$ is called sofic if it admits a sofic approximation: a sequence of asymptotically free almost actions on finite sets. Given a sofic group $G$, it is a natural problem to try to classify all its sofic approximations and, more generally, its asymptotic homomorphisms to finite symmetric groups. Ideally, one would aim to show that any almost homomorphism from $G$ to a finite symmetric group is close to an actual homomorphism. If this is the case, then $G$ is called stable in permutations, or P-stable. In this talk, I will first present a result providing a large class of product groups are not P-stable. In particular, the direct products of the free group on two generators with itself and with the group of integers are not P-stable. This implies that P-stability is not closed under the direct product construction, which answers a question of Becker, Lubotzky and Thom. I will also present a more recent result, which strengthens the above in the case when $G$ is the direct product of the free group on two generators with itself. This shows, answering a question of Bowen, that $G$ admits a sofic approximation which is not essentially a “branched cover” of a sofic approximation by homomorphisms.