2/28/2006

Alex Lubotzky
Institute for Advanced Study and Hebrew University

Finite groups and hyperbolic manifolds

The isometry group of a closed hyperbolic n-manifold is finite. We prove that for every n>1 and every finite group G there is an n-dimensional closed hyperbolic manifold whose isometry group is G. This resolves a long standing problem whose low dimensional cases n=2 and n=3 were proved by Greenberg ('74) and Kojima ('88) resp. The proof is nonconstructive; it uses a 'probabilistic method', i.e. counting results from the theory of 'subgroup growth'. The talk won't assume any prior knowledge on the subject. Based on joint work with M. Belolipetsky (Inven. Math. Dec. 2005)