Volodymyr Nekrashevych Texas A&M University - College Station We will discuss exponential estimates of divergence of geodesics in Cayley graphs of hyperbolic groupoids, and natural metrics on their unit spaces. As applications we will describe a generalization for Smale spaces of a theorem of M.Brin and A. Manning about Anosov diffeomorphisms. We will also discuss a general approach to construction of bounded Fredholm modules associated with hyperbolic dynamical systems.