The quasiconformal equivalence of Riemann surfaces and the universal Schottky space

TL;DR

This paper explores the quasiconformal equivalence of Riemann surfaces, especially of infinite type, and introduces geometric conditions for equivalence, culminating in the construction of the universal Schottky space that encompasses all Schottky spaces.

Contribution

It provides new geometric criteria for quasiconformal equivalence of Riemann surfaces of infinite type and constructs the universal Schottky space as a unifying framework.

Findings

Constructed an example illustrating the complexity of quasiconformal equivalence for infinite type surfaces.

Established geometric conditions for quasiconformal equivalence.

Defined the universal Schottky space containing all Schottky spaces.

Abstract

In the theory of Teichm\"uller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In this paper, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to obtain the universal Schottky space which contains all Schottky spaces, the deformation spaces of Schottky groups as the universal Teichm\"uller space contains all Teichm\"uller spaces.