Finite Groups and Hyperbolic Manifolds

M. Belolipetsky, A. Lubotzky

Published 2004-06-29, updated 2005-01-19Version 2

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.