On quasiconformal equivalence of Schottky regions

Abstract

This paper concerns the general question: When are two homeomorphic Riemann surfaces quasiconformally equivalent? For the case of finite-type Riemann surfaces, the answer is known; it depends on the genus, the number of punctures, and the number of hyperbolic boundaries. On the other hand, for the case of infinite-type Riemann surfaces, this is rather complicated. We provide a general solution for those Riemann surfaces $S=\hat{\mathbb{C}}\setminus \Lambda$, where $\Lambda$ is a Cantor set being the limit set of a finitely generated Kleinian group. As a consequence, we obtain that there are four different Teichmüller spaces of Cantor limit sets of finitely generated Kleinian groups.