Products of twists, geodesic-lengths and Thurston shears

TL;DR

This paper explores the symplectic geometry and algebraic structures related to shear deformations on geodesic laminations of hyperbolic surfaces with cusps, establishing dualities and algebraic equivalences.

Contribution

It introduces a new framework linking shear deformations, geodesic lengths, and Thurston shears, and proves algebraic and geometric dualities in this context.

Findings

Weil-Petersson duality of shears and lengths established.

Equality of Fock shear coordinate algebra and WP Poisson algebra proven.

Explicit formula for WP Riemannian pairing of shears provided.

Abstract

Thurston introduced shear deformations (cataclysms) on geodesic laminations - deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined and the Weil-Petersson (WP) duality of shears and the defined length is established. The Poisson bracket of a pair of balanced weight systems on a set of disjoint ideal geodesics is given in terms of an elementary 2-form. The symplectic geometry of balanced weight systems on ideal geodesics is developed. Equality of the Fock shear coordinate algebra and the WP Poisson algebra is established. The formula for the WP Riemannian pairing of shears is also presented.