Tessellating the moduli space of strictly convex projective structures on the once-punctured torus

TL;DR

This paper constructs a natural cell decomposition of the moduli space of finite-volume, strictly convex real projective structures on the once-punctured torus using Euclidean cell decompositions, coordinates, and algebraic geometry.

Contribution

It introduces a new cell decomposition of the moduli space based on Euclidean decompositions and algebraic geometry techniques, extending to the thrice-punctured sphere.

Findings

Moduli space admits a natural cell decomposition.

Coordinates by Fock and Goncharov facilitate the construction.

Cell decomposition extends to the thrice-punctured sphere.

Abstract

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of framed strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.