Growth of the Weil-Petersson Diameter of Moduli Space

Details Growth of the Weil-Petersson Diameter of Moduli Space Growth of the Weil-Petersson Diameter of Moduli Space

2010-04-18

arxiv.org/abs/1004.3029v2

Mathematics

Geometric Topology

26 pages, 7 figures

Scientific paper

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.

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