Finite Groups and Hyperbolic Manifolds
Group Theory
2009-11-10 v2 Geometric Topology
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.
Cite
@article{arxiv.math/0406607,
title = {Finite Groups and Hyperbolic Manifolds},
author = {M. Belolipetsky and A. Lubotzky},
journal= {arXiv preprint arXiv:math/0406607},
year = {2009}
}
Comments
12 pages, to appear in Invent. Math