Finite groups and hyperbolic manifolds

@article{Belolipetsky2005FiniteGA,
  title={Finite groups and hyperbolic manifolds},
  author={Mikhail V. Belolipetsky and Alexander Lubotzky},
  journal={Inventiones mathematicae},
  year={2005},
  volume={162},
  pages={459-472}
}
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 (1974) and Kojima (1988), respectively. Our proof is non constructive: it uses counting results from subgroup growth theory to show that such manifolds exist. 

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