Abstract

In recent years there has been a considerable interest in questions regarding approximate homomorphisms between groups, going under the name of group stability. For our setting, a group G is said to be HS-stable if any (pointwise) approximate finite dimensional unitary representation of G is (pointwise) close to a true unitary representation of G, where proximity is measured by the normalized Hilbert-Schmidt norm. In this talk we will introduce and connect stability to other important questions in approximation theory and representation theory of infinite discrete groups. In the situation of amenable groups, stability can be translated to a question regarding character theory of the group, yielding a useful criterion for giving examples. In particular, it can be utilized to prove existence of uncountably many such groups (joint work with Vigdorovich and Levit). In the opposite case of property (T) groups, such as lattices in simple higher rank Lie groups and Gromov random groups, stability is a potentially rare property. Indeed, we showed that in many cases, weakenings of stability would imply the existence of a non-sofic group.