Growth of the Weil-Petersson Diameter of Moduli Space

TL;DR

This paper investigates how the diameter of the compactified moduli space of Riemann surfaces grows with genus and number of marked points, providing bounds and growth rates in terms of these parameters.

Contribution

It establishes new bounds on the growth of the Weil-Petersson diameter of moduli space as functions of genus and marked points.

Findings

Diameter grows as √n in n

Diameter is bounded above by C√g log g in g

Provides lower bounds related to the thick part 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.