  The isometry group of a compact n-dimensional hyperbolic manifold is known to
be finite. We show that for every n > 2, every finite group is realized as the
full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and
n = 3 have been proven by Greenberg and Kojima, respectively. Our proof is non
constructive: it uses counting results from subgroup growth theory and the
strong approximation theorem to show that such manifolds exist.
