The quasiconformal equivalence of Riemann surfaces and the universal Schottky space

Shiga, Hiroshige

doi:10.1090/ecgd/343

The quasiconformal equivalence of Riemann surfaces and the universal Schottky space
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by Hiroshige Shiga

Conform. Geom. Dyn. 23 (2019), 189-204

DOI: https://doi.org/10.1090/ecgd/343

Published electronically: October 30, 2019

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Abstract:

In the theory of Teichmüller 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 a universal property of the deformation spaces of Schottky regions, which is analogous to the fact that the universal Teichmüller space contains all Teichmüller spaces.

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