A dynamical perspective on shear-bend coordinates

Abstract

Twisted ${\mathrm{SL}}_2\mathbb{C}$ local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface with punctures, Gaiotto, Hollands, Moore and Neitzke's "abelianization" process computes the shear-bend coordinates of a twisted ${\mathrm{SL}}_2\mathbb{C}$ local system without reference to its hyperbolic geometry. Using tools from dynamics, we'll generalize abelianization to compact surfaces, leading to a dynamical recipe for the shear-bend parameterization. This recipe lends itself well to numerical approximation, and it may clarify the changes of coordinates that relate different shear parameterizations.

arXiv comment100 pages, 37 figures. v2: added secondary result (theorem 1.2.B); greatly simplified main argument, strengthening main result (theorem 1.2.A); cut material which is no longer needed (but still available in reference [1]); corrected argument in section 8.3.1 (conclusion unaffected); slightly corrected review in section 3.2.3

References in corpus (7)

• Moduli spaces of local systems and higher Teichmuller theory
• Hitchin characters and geodesic laminations
• Spectral networks and Fenchel-Nielsen coordinates
• Geometry of Character Varieties of Surface Groups
• Khinchin type condition for translation surfaces and asymptotic laws for the Teichmuller flow
• Thurston's cataclysms for Anosov representations
• Khinchin theorem for interval exchange transformations

Cited by in corpus (2)

• Mirzakhani's work on earthquake flow
• Spectral Networks and Non-abelianization