In this talk we consider smooth manifolds N that are homeomorphic but not diffeomorphic to a hyperbolic manifold M, and ask "How much symmetry does N have? For example, does Isom(M) act on N by diffeomorphisms?" In general the answer is "No" by work of Farrell-Jones. We will discuss this problem, explain its relation to Nielsen realization, and give a generalization of the Farrell-Jones result. This is joint work with Mauricio Bustamante. Speaker(s): Bena Tshishiku (Brown University)