Abstract. We exhibit strong constraints on the geometry and topology of a uniformly quasiconformally homogeneous hyperbolic manifold. In particular, if n ≥ 3, a hyperbolic n-manifold is uniformly quasiconformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold. Moreover, if n ≥ 3, we show that there is a constant K n > 1 such that if M is a hyperbolic n-manifold, other than H n , which is K-quasiconformally homogeneous, then K ≥ K n .

We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff dimension, that extends the Weil-Petersson metric on Teichmüller space. We use this to describe a metric on Teichmüller space obtained by taking the second derivative of Hausdorff dimension and show that this metric is bounded below by the Weil-Petersson metric. We relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil-Petersson metric of the associated earthquake vector of the lamination.

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Λ(M ) be the supremum of λ 0 (N ) where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of M . Similarly, we let D(M ) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M . We observe that Λ(M ) = D(M )(2 − D(M )) if M is not handlebody or a thickened torus. We characterize exactly when Λ(M ) = 1 and D(M ) = 1 in terms of the characteristic submanifold of the incompressible core of M .

Abstract. We show that any closed hyperbolic surface admitting a conformal automorphism with "many" fixed points is uniformly quasiconformally homogeneous, with constant uniformly bounded away from 1. In particular, there is a uniform lower bound on the quasiconformal homogeneity constant for all hyperelliptic surfaces. In addition, we introduce more restrictive notions of quasiconformal homogeneity and bound the associated quasiconformal homogeneity constants uniformly away from 1 for all hyperbolic surfaces.

No abstract