Concrete construction of cone hyperbolic structures for a 3-dimensional cone manifold with non-compact cone singularity

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The space of complete cone hyperbolic structures on the 3-dimensional cone manifold obtained as the product of the torus with a single cone point and the interval has been studied, where a conjecture of Akiyoshi and Yamashita, which expects a certain relation between geometry and algebra, has been focused on. The following facts has been revealed: (1) The conjecture is true on the subspace consisting of real representations. (2) The simple loops are realized by geodesics for the cone hyperbolic structure associated with "good fundamental polyhedron". Moreover, certain bound on the lengths of geodesics are found. (3) The notion of Ford domain is generalized to the complete cone hyperbolic structures containing a "standard horospherical neighborhood", and the good fundamental polyhedra are shown to be the Ford domains.