Analysis of structures of infinite dimensional Teichmuller spaces and complex analytic moduli spaces

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For research of the infinite dimensional Teichmuller space of an analytically infinite Riemann surface, we consider the asymptotic Teichmuller space which is a quotient space of the Teichmuller space. A quasiconformal mapping class of a Riemann surface acts on the asymptotic Teichmuller space biholomorphically as an asymptotic Teichmuller modular transformation, but it can act trivially, which is different from the case of the Teichmuller space. In this research, we gave a characterization of the asymptotically trivial mapping class group. Furthermore, we proved the fixed point theorem for the asymptotic Teichmuller modular group and gave an answer to the asymptotic version of the Nielsen realization problem.