  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 \pi_0\Diff_0(N) is infinitely generated with similar
results holding topologically.
