Abstract

The notion of group stability, which asks if an “almost” homormorphism is “close” to a homomorphism, was introduced in Dogon’s talk last week, and studied in the setting of pointwise Hilbert-Schmidt stability. In this talk, we will explore the question in the context of uniform distance with respect to the operator norm. This is a more natural question that goes all the way back to Turing and Ulam, with subsequent developments by Kazhdan, et al. The main tool we use is a non-standard asymptotic variant of bounded cohomology of groups, devised so that the vanishing of this cohomology implies stability in this setting. 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.