Finite Groups and Hyperbolic Manifolds

  title={Finite Groups and Hyperbolic Manifolds},
  author={Alexander Lubotzky},
The isometry group of a compact n-dimensional hyperbolic man-ifold 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 [G] and Ko-jima [K], respectively. Our proof is non constructive: it uses counting results from subgroup growth theory to show that such manifolds exist. 

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