Abstract
We consider the infinite dimensional Teichmuller space of a Riemann surface of general type. On the basis of the fact that the action of the quasiconformal mapping class group on the Teichmuller space is not discontinuous, in general, we divide the Teichmuller space into two disjoint subsets, the limit set and the region of discontinuity, according to the discreteness of the orbit by a subgroup of the quasiconformal mapping class group. The asymptotic Teichmuller space is a certain quotient space of the Teichmuller space and there is a natural projection from the Teichmuller space to the asymptotic Teichmuller space. We consider the fibers of the projection over any point in the asymptotic Teichmuller space, and show a coherence of the discreteness on each fiber in the Teichmuller space.
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