Title: "Almost" Representations, Group Stability and Cohomology

Abstract: Consider the following natural robustness question: is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This classical question of stability goes all the way back to Turing and Ulam, and can be posed for different target groups, and different notions of distance. Group stability has been an active line of study in recent years, thanks to its connections to major open problems like the existence of non-sofic and non-hyperlinear groups, the group Connes embedding problem and the recent breakthrough result MIP*=RE, apart from property testing and error-correcting codes.

In this talk, we will survey some of the main results and questions in this area, with a focus on stability in the uniform setting and distances given by submultiplicative norms. The main tool we use is a new non-standard asymptotic variant of bounded cohomology of groups, devised so that the vanishing of this cohomology implies stability in this setting (an almost homomorphism would correspond to a 2-cocycle, and it being a coboundary would correspond to a correction to a homomorphism). 

We will then sketch how, apart from unifying earlier stability results, this theory can help prove stability for a large class of groups, including high-rank lattices and lamplighters.

Based on joint works with Glebsky, Lubotzky, Monod, and Fournier-Facio.