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

Abstract

Let $T_0$ be the once-punctured torus and θ a real number with $0<\theta <2\pi$. Let $\tilde{\rho }$ be a representation of ${\pi }_1(T_0)$ to $\mathit{SL}(2,\mathbb{R})$ which sends the peripheral loop to an elliptic element with trace $-2\cos (\theta /2)$. Let ρ be the $\mathit{PSL}(2,\mathbb{R})$-representation induced from $\tilde{\rho }$, 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 $\mathbb{R}$ which induces ρ as the holonomy representation.