The complex-symplectic geometry of SL(2,C)-characters over surfaces

Abstract

The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M determine regular functions which generate complex Hamiltonian flows. For simple closed curves, these complex Hamiltonian flows arise from holomorphic flows on the representation variety generalizing the Fenchel-Nielsen twist flows on Teichmueller space and the complex quakebend flows on quasi-Fuchsian space. Closed curves in the complex trajectories of these flows lift to paths in the deformation space of complex-projective structures between different projective structures with the same holonomy (grafting). A pants decomposition determines a holomorphic completely integrable system on X. This integrable system is related to the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space developed by Tan and Kourouniotis, and relate to recent formulas of Platis and Series on complex-length functions and complex twist flows.

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• Delambre-Gauss Formulas for Augmented, Right-Angled Hexagons in Hyperbolic 4-Space
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• Families of Hitchin systems and N=2 theories
• Automorphisms of the Fricke characters of groups
• $S{L}_2(\mathbb{C})$-Character Variety of a Hyperbolic Link and Regulator
• Minimal surfaces and symplectic structures of moduli spaces
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