On the Automorphism Groups of Hyperbolic Manifolds

Abstract Let ${\mathrm{Diff}}_{0}(N)$ represent the subgroup of diffeomorphisms that are homotopic to the identity. We show that if $N$ is a closed hyperbolic 4-manifold, then $\pi _{0}{\mathrm{Diff}}_{0}(N)$ is not finitely generated with similar results holding topologically. This proves in dimension-4 results previously known for $n$-dimensional hyperbolic manifolds of dimension $n\ge 11$ by Farrell and Jones in 1989 and $n\ge 10$ by Farrell and Ontaneda in 2010. Our proof relies on the technical result that $\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$ is not finitely generated, which extends to the topological category smooth results of the authors. We also show that $\pi _{n-4} {\mathrm{Homeo}}(S^{1} \times D^{n-1})$ is not finitely generated for $n \geq 4$ and in particular $\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$ is not finitely generated. These results are new for $n=4, 5$ and $7$. We also introduce higher dimensional barbell maps and establish some of their basic properties.