Abstract

Ergodic theory has proven to be a powerful tool for studying negatively curved Riemannian manifolds and their fundamental groups. For instance, the seminal work of Margulis extracts from the mixing of the geodesic flow precise asymptotics for the number of closed geodesics of a given length. This approach has been extended beyond the realm of manifolds, e.g., to CAT spaces and CAT spaces with a rank-one element. This talk will report on an ongoing project whose goal is to carry (some of) these tools to the context of geometric group theory: given a general metric space X with a weak form of negative curvature (captured by the existence of a contracting element in its isometry group), we will explain how to build a geodesic flow on X and investigate its dynamical properties, such as ergodicity and mixing. joint work with Samuel Tapie