Finite groups and hyperbolic manifolds

By Mikhail V. Belolipetsky, Alexander Lubotzky. 2004

57 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a3938b7f6528fcc4294008cef321a6ff830ba4d5

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Finite covers of random 3-manifolds (2005, methodology): “[Zuk, Gro2]. Very recently, Belolipetsky and˙ Lubotzky have used random techniques to show that given n and a ﬁnite group G there exists a hyperbolic n-manifold whose full isometry group is exactly G [BL]. Perhaps similar techniques could be applied to questions about 3-manifolds. In particular, to ”
Systoles of hyperbolic manifolds (2010, methodology): “This enables one to use our manifolds for the construction of Belolipetsky and Lubotzky [3], which proves that every finite group can be realised as the full isometry group of some compact hyperbolic n–manifold.”