On the Automorphism Groups of Hyperbolic Manifolds

Let represent the subgroup of diffeomorphisms that are homotopic to the identity. We show that if is a closed hyperbolic 4-manifold, then is not finitely generated with similar results holding topologically. This proves in dimension-4 results previously known for -dimensional hyperbolic manifolds of dimension by Farrell and Jones in 1989 and by Farrell and Ontaneda in 2010. Our proof relies on the technical result that is not finitely generated, which extends to the topological category smooth results of the authors. We also show that is not finitely generated for and in particular is not finitely generated. These results are new for and. We also introduce higher dimensional barbell maps and establish some of their basic properties.