Growth of the Weil–Petersson diameter of moduli space

Abstract

In this paper we study the Weil–Petersson geometry of $\overline{{ℳ}_{g,n}}$, 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 $\overline{{ℳ}_{g,n}}$ as a function of $g$ and $n$. We show that this diameter grows as $\sqrt{n}$ in $n$, and is bounded above by $C\sqrt{g}logg$ in $g$ for some constant $C$. We also give a lower bound on the growth in $g$ of the diameter of $\overline{{ℳ}_{g,n}}$ in terms of an auxiliary function that measures the extent to which the thick part of moduli space admits radial coordinates.