The symplectic nature of fundamental groups of surfaces. (English) Zbl 0574.32032

A natural symplectic structure on the space of representations \(\pi\) \(\to G\), where \(\pi\) is the fundamental group of a Riemann surface and G is a semisimple Lie group, is introduced. The author considers this result as a common cause for the existence of a symplectic structure on different moduli spaces associated with Riemann surfaces. For example if \(G=PSL_ 2({\mathbb{R}})\) then the symplectic structure coincides with the Weil- Petersson’s one on Teichmüller space.
Reviewer: A.Givental’


32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H30 Coverings of curves, fundamental group
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
30F99 Riemann surfaces
14E20 Coverings in algebraic geometry
32G13 Complex-analytic moduli problems
22E46 Semisimple Lie groups and their representations
14H15 Families, moduli of curves (analytic)
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