A group is called stable if its almost homomorphisms into finite-dimensional unitary groups are close to actual homomorphisms, where proximity is measured in the normalized trace norm. I will discuss two works which rely on the character-theoretic criterion of Hadwin and Shulman, resulting in plenty of new examples.

With Dogon and Levit we show that the uncountable family of 2-generated groups constructed by B.H. Neumann are all stable. The proof uses results in character theory of finite symmetric groups.

With Levit we find a strong connection between stability of groups and stability and approximation properties of dynamical systems. We use this to establish stability of various groups, among them are all virtually nilpotent groups and lamplighter groups.