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

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.

arXiv comment34 pages, 6 figures. Version before submission. Comments are welcome

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Cited by in corpus (1)

• McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential