Growth of the Weil-Petersson Diameter of Moduli Space

Abstract

In this paper we study the Weil-Petersson geometry of $\overline{{\mathcal{M}}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the diameter of $\overline{{\mathcal{M}}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C\sqrt{g}\log g$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{{\mathcal{M}}_{g,n}}$ in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates.

26 pages, 7 figures

References in corpus (2)

• Diastolic and isoperimetric inequalities on surfaces
• Estimates of Weil-Petersson volumes via effective divisors

Cited by in corpus (7)

• Growth of the Weil-Petersson inradius of moduli space
• Diameter of the thick part of moduli space and simultaneous Whitehead moves
• Small eigenvalues of closed Riemann surfaces for large genus
• Uniform Bounds for Weil-Petersson Curvatures
• A new uniform lower bound on Weil-Petersson distance
• Systole functions and Weil-Petersson geometry
• On the Weil-Petersson curvature of the moduli space of Riemann surfaces of large genus