On the automorphism groups of hyperbolic manifolds

Abstract

Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the n-manifold N, pointwise. We show that the (n-4)-th homotopy group of Homeo(S^1 \times D^{n-1}) is not finitely-generated for n >= 4 and in particular the topological mapping-class group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (when n >= 4) do not have the homotopy-type of finite CW-complexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic n-manifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n-4)-th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then π_0\Diff_0(N) is infinitely generated with similar results holding topologically.