Fock and Goncharov Moduli Space and its Poisson Structure

Abstract: Vladimir Fock and Alexander Goncharov recently (2006) introduced a very convenient coordinate system to describe $T_3^{+}(S)$, the moduli space of framed convex projective structures with geodesic boundary on a surface $S$. Via such coordinate system, they defined a Poisson structure on $T_3^{+}(S)$ closely related to Goldman’s Poisson structure introduced in 1984. We analysed Fock and Goncharov’s parametrisation and gave a geometric proof of the equivalence between the two Poisson structures. I will introduce F&G’s coordinate system and Poisson structure, and outline our idea in the proof. This work is part of a joint project with R. Haraway, D. Tate and S. Tillmann.