Earthquakes on the once-punctured torus

TL;DR

This paper explores explicit methods to describe earthquake deformations on the Teichmüller space of the once-punctured torus, linking algebraic recurrence relations with hyperbolic geometry insights.

Contribution

It introduces two complementary methods for explicit earthquake deformation calculations and analyzes their geometric and algebraic interpretations.

Findings

Methods align, confirming consistency between algebraic and geometric approaches.

Expressions are transformed into various coordinate systems for deeper analysis.

Examined limiting behavior offers insights into earthquakes about measured geodesic laminations.

Abstract

We study earthquake deformations on Teichm\"uller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichm\"uller space to examine earthquake deformations further. Two families of curves are used as examples. Examining the limiting behaviour of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.