The Goldman and Fock-Goncharov coordinates for convex projective structures on surfaces

TL;DR

This paper compares two coordinate systems for the space of convex projective structures on surfaces, explicitly describing the transformation between them, with detailed focus on pairs of pants.

Contribution

It explicitly describes the coordinate change between Goldman and Bonahon-Dreyer (Fock-Goncharov) coordinates for convex projective structures on surfaces.

Findings

Explicit coordinate change formulas derived

Focus on pairs of pants for detailed analysis

Bridges two different parametrizations of the moduli space

Abstract

Let P(S) be the space of convex projective structures on a surface S with negative Euler characteristic. Goldman and Bonahon-Dreyer constructed two different sets of global coordinates for P(S), both associated to a pair of pants decomposition of the surface S. The article explicitly describes the coordinate change between these two parametrizations. Most of the arguments are concentrated in the case where S is a pair of pants, in which case the Bonahon-Dreyer coordinates are actually due to Fock-Goncharov.