Groups acting at infinity

Abstract

An action of a group on a topological space is rigid if small perturbations to the action have little meaningful influence on its global dynamics. Many examples of rigid actions come from geometric considerations. This introductory survey describes the idea of “looking to infinity” as a source both of rigid examples and proofs of rigidity, starting with some early history then passing quickly to recent developments in topological rigidity of group actions. The examples considered include actions of hyperbolic manifold groups on the visual boundary of their universal cover, automorphism groups of surface groups, boundary actions of hyperbolic groups in the sense of Gromov, and group actions derived from Anosov flows on 3-manifolds.