On a Closed Binding Curve of One-holed Torus

TL;DR

This paper investigates the geometric structures of one-holed tori with a specific binding curve, revealing new coordinate systems for their Fricke space through polygon decompositions.

Contribution

It introduces a novel coordinate system for the Fricke space of the one-holed torus based on polygon decompositions related to a specific binding curve.

Findings

Decomposition into polygons with geodesic sides for hyperbolic structures.

Existence of equilateral bigons with punctures and hexagons with equal sides.

New parametrization of the Fricke space for the one-holed torus.

Abstract

Given a closed binding curve $\gamma$ of a surface $\Sigma$, any equivalence class of marked complete hyperbolic structure can be decomposed into polygons(possibly with a puncture) with sides being hyperbolic geodesic segments. When $\Sigma$ is a one-holed torus and $\gamma = A^3 B^2$, we show that any equivalence class of marked complete hyperbolic structure gives rise to an equilateral bigon with a puncture and a hexagon with equal opposite sides. In particular, we give a new coordinates of the Fricke Space of the one-holed torus.