Growth of the Weil-Petersson Diameter of Moduli Space

William Cavendish; Hugo Parlier

doi:10.1215/00127094-1507312

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arXiv · 1004.3029

Growth of the Weil-Petersson Diameter of Moduli Space

William Cavendish·Hugo Parlier

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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.

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BibTeX RIS

William Cavendish, Hugo Parlier. 2010-10-13. Growth of the Weil-Petersson Diameter of Moduli Space. https://doi.org/10.1215/00127094-1507312

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