On Moduli Spaces of Convex Projective Structures on Surfaces: Outitude and Cell-Decomposition in Fock-Goncharov Coordinates

TL;DR

This paper extends the canonical cell decomposition from hyperbolic to convex projective structures on punctured surfaces using Fock-Goncharov coordinates, providing an algorithm for cell determination and exploring semi-arithmetic holonomy groups.

Contribution

It generalizes the Epstein-Penner decomposition to convex projective structures and introduces an edge-flipping algorithm for cell determination in moduli space.

Findings

Extended cell decomposition to convex projective structures.

Developed an intrinsic edge-flipping algorithm.

Identified semi-arithmetic holonomy groups in many cases.

Abstract

Generalising a seminal result of Epstein and Penner for cusped hyperbolic manifolds, Cooper and Long showed that each decorated strictly convex projective cusped manifold has a canonical cell decomposition. Penner used the former result to describe a natural cell decomposition of decorated Teichm\"uller space of punctured surfaces. We extend this cell decomposition to the moduli space of decorated strictly convex projective structures of finite volume on punctured surfaces. The proof uses Fock and Goncharov's $\mathcal{A}$-coordinates for doubly decorated structures. In addition, we describe a simple, intrinsic edge-flipping algorithm to determine the canonical cell decomposition associated to a point in moduli space, and show that Penner's centres of Teichm\"uller cells are also natural centres of the cells in moduli space. We show that in many cases, the associated holonomy groups are semi-arithmetic.