Finite Groups and Hyperbolic Manifolds

Abstract

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.

12 pages, to appear in Invent. Math

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