Exotic quasi-conformally homogeneous surfaces

Abstract

Abstract. We construct uniformly quasiconformally homogeneous Riemann surfaces which are not quasiconformal deformations of regular covers of closed orbifolds.

The critical exponent, the Hausdorff dimension of the limit set and the convex core entropy of a Kleinian group

“…Since the number of such permutations is finite, there is a uniform constant K ≥ 1 such that the modified map f can be chosen as a K-quasi-isometry on each Q. We can find similar arguments in Bonfert-Taylor et al [12]. Since the action on G induced by f is isometric, we see that such f is a rough-isometry with an additive constant L depending only on S and K. Lifting f to B 2 , we obtain the required rough-isometry f .…”

Quasiconformal homogeneity after Gehring and Palks

“…One might then optimistically hope that every uniformly quasiconformally homogeneous surface is a quasiconformal deformation of a regular cover of a closed hyperbolic 2-orbifold. Bonfert-Taylor, Canary, Souto and Taylor [7] showed that this is not the case. Sketch of proof: Given a connected countable graph X, each of whose vertices has valence d ≥ 3, one may construct a hyperbolic surface S X by "thickening up" X.…”

There are no exotic ladder surfaces

“…QCH surfaces that are not quasiconformally equivalent to regular covers of closed orbifolds) is shown. However, all the exotic QCH surfaces constructed in [4] are homeomorphic; in particular, they are homeomorphic to the one-ended infinite-genus surface (affectionately referred to as the Loch Ness monster surface).…”