Expansion in groups (or their Cayley graphs) is a valuable and well-studied notion in both mathematics and computer science, and describes a robust form of connectivity of graphs (a gap property of fixed points of representations of groups). It can also be interpreted as a graph on which connectivity is efficiently locally testable.

Group stability, on the other hand, is concerned with another robustness property – but of homomorphisms (or representations). Namely, is an almost-homomorphism of a group necessarily a small deformation of a homomorphism? This too can be interpreted as a local testability property of group homomorphisms in the right settings.

Expansion in groups (or property (T)) had been classically reformulated in the language of algebraic topology – in terms of the vanishing of the first cohomology of the group. In this talk we will see approaches in capturing group stability in terms of the vanishing of a second cohomology of the group, motivating higher-dimensional generalizations of expansion.

Based on joint (previous and ongoing) works with Monod, Glebsky, Lubotzky, Fournier-Facio, Dogon.