Canonical polygons for the hyperbolic structures on the torus with a single cone point

Abstract

Let T0 be the once-punctured torus and θ a real number with 0<θ<2π. Let ρ˜ be a representation of π1(T0) to SL(2,R) which sends the peripheral loop to an elliptic element with trace −2cos⁡(θ/2). Let ρ be the PSL(2,R)-representation induced from ρ˜, and assume it satisfies Bowditch's Q-condition. In this paper, we construct a certain polyhedron, which is obtained as a variation of Jorgensen's theory to cone manifolds, and construct a complete cone hyperbolic structure on the 3-dimensional cone manifold obtained as the product of the torus with a single cone point and R which induces ρ as the holonomy representation.