Quasiconformal homogeneity of hyperbolic manifolds

We exhibit strong constraints on the geometry and topology of a uniformly quasiconformally homogeneous hyperbolic manifold. In particular, if n3, a hyperbolic n-manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold. Moreover, if n3, we show that there is a constant K_n>1 such that if M is a hyperbolic n-manifold, other than which is K–quasiconformally homogeneous, then KK_n.Mathematics Subject Classification (2000): 30C60Research supported in part by NSF grant 070335 and 0305704.Research supported in part by NSF grant 0203698.Research supported in part by the NZ Marsden Fund and the Royal Society (NZ).Research supported in part by NSF grant 0305704.