Moduli spaces of real projective structures on surfaces: Notes on a paper by V.V. Fock and A.B. Goncharov

TL;DR

This paper provides a comprehensive overview of Fock and Goncharov's methods for describing moduli spaces of real projective structures on surfaces, including their Poisson structures and connections to character varieties.

Contribution

It offers a self-contained exposition of Fock and Goncharov's description of moduli spaces and derives related results, clarifying the structure and properties of these spaces.

Findings

Description of moduli space of convex projective structures

Relationship between Poisson structures and character varieties

Derivation of Marquis and Goldman's results

Abstract

These notes grew out of our learning and applying the methods of Fock and Goncharov concerning moduli spaces of real projective structures on surfaces with ideal triangulations. We give a self-contained treatment of Fock and Goncharov's description of the moduli space of framed marked properly convex projective structures with minimal or maximal ends, and deduce results of Marquis and Goldman as consequences. We also discuss the Poisson structure on moduli space and its relationship to Goldman's Poisson structure on the character variety.