Volume of the moduli space of unmarked bounded positive convex $\mathbb{RP}^2$ structures

TL;DR

This paper studies the volume of moduli spaces of convex real projective structures on surfaces, demonstrating finiteness of volume for certain bounded subsets and establishing analogs of Mumford's compactness theorem.

Contribution

It provides bounds on the Goldman symplectic volume of subsets with bounded projective invariants and proves a Mumford-type compactness result for area-bounded subsets.

Findings

Volume of bounded subsets is polynomially bounded in parameters.

All subsets with bounded invariants have finite volume.

Mumford's compactness theorem analog holds for area-bounded subsets.

Abstract

For the moduli space of unmarked convex $\mathbb{RP}^2$ structures on the surface $S_{g,m}$ with negative Euler characteristic, we investigate the subsets of the moduli space defined by the notions like boundedness of projective invariants, area, Gromov hyperbolicity constant, quasisymmetricity constant etc. These subsets are comparable to each other. We show that the Goldman symplectic volume of the subset with certain projective invariants bounded above by $t$ and fixed boundary simple root lengths $\mathbf{L}$ is bounded above by a positive polynomial of $(t,\mathbf{L})$ and thus the volume of all the other subsets are finite. We show that the analog of Mumford's compactness theorem holds for the area bounded subset.