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

The quasiconformal equivalence of Riemann surfaces and the universal Schottky space