Study of metric properties of Teichmuller spaces

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We have studied metric properties of Teichmuller space of 2 dimensional hyperbolic orbifolds and related subjects on discontinuous groups and 2 and 3 dimensional orbifolds. We obtained the following results :1. Description of Teichmuller space of hyperbolic cone surfaces as a real algebraic space by using the distances between cone points and horocycles and its applications to the problem of finding the minimal number of geodesic length functions needed for a global parametrization of a Teichmuller space, the representation of mapping class groups and holomorphic families of Riemann surfaces.2. Geodesic length functions related to suitably chosen closed curves on the underlying surface of a cone surface supply a global coordinate-system on the Teichmuller space. We expressed the 2 form which is defined analogously to Wolpert's formula of Weil-Petersson 2-form in terms of these geodesic length functions and calculated the volumes of some moduli spaces.3. Classification of all non-cocompact arithmetic. Fuchsian groups of signature (0 ; theta_1, theta_2, theta_3, theta_4).