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Products of twists, geodesic lengths and Thurston shears

Part of: Deformations of analytic structures Other groups of matrices Symplectic geometry, contact geometry Riemann surfaces

Published online by Cambridge University Press:  09 October 2014

Scott A. Wolpert
Scott A. Wolpert*
Affiliation:
University of Maryland, College Park, MD 20742-4015, USA email [email protected]
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Abstract

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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.

Keywords

Teichmüller spaceWeil–Petersson geometrygeodesic-length functionsshear and twist deformationsFock shear coordinate algebra

MSC classification

Primary: 30F60: Teichmüller theory
Secondary: 20H10: Fuchsian groups and their generalizations 32G15: Moduli of Riemann surfaces, Teichmüller theory 53D30: Symplectic structures of moduli spaces
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 2 , February 2015 , pp. 313 - 350
Copyright
© The Author 2014 

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