Analysis of deformation space of 3-dimensional cone-hyperbolic structures using fundamental domains arising from cut loci

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The space of cone-hyperbolic structures for the 3-dimensional cone-manifold obtained as the product of the torus with a cone-point and the real line was studied in order to establish the deformation theory of 3-dimensional cone-hyperbolic structures for cone-manifolds with noncompact cone singularity. The deformation was studied by using a variant of Ford domains in the theory of Kleinian groups. A geometric parametrization for such space was established, which is conjectured to give a parametrization for certain slice of the character variety of one-holed torus defined by Tan et al. in terms of dynamics on the variety.