Thurston's cataclysms for Anosov representations

2013/01/29 by Guillaume Dreyer, Dreyer, Guillaume · 1 citation
Mathematics · #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.1301.6961

Abstract

Given an Anosov representation ρ\colon π₁(S) → \PSL_n(ℝ) and a maximal geodesic lamination λ in a surface S, we construct shear deformations along the leaves of the geodesic lamination λ endowed with a certain flag decoration, that is provided by the associated flag curve F_ρ\colon \Sinf → Flag(ℝⁿ) of the Anosov representation ρ; these deformations generalize to Labourie's Anosov representations Thurston's cataclysms for hyperbolic structures on surfaces. A cataclysm is parametrized by a transverse n--twisted cocycle for the orientation cover \La of λ. In addition, we establish various geometric properties for these deformations. Among others, we prove a variation formula for the associated length functions ℓⁱ_ρ of the Anosov representation ρ.

Cited by

• A dynamical perspective on shear-bend coordinates

Related

• d-pleated surfaces and their shear-bend coordinates
• d‐Pleated surfaces and their shear‐bend coordinates
• Anosov deformations of Barbot representations
• Anosov Flows, Surface Groups and Curves in Projective Space
• Geodesic length spectrum of hyperelliptic connected components