The Weil-Petersson diameter of moduli space for large genus

Abstract

We determine the large-genus order of the Weil-Petersson diameter of ${\mathcal{M}}_{g,n}$ for each fixed number of punctures: namely, $\mathrm{diam}({\mathcal{M}}_{g,n})\asymp \sqrt{g}\log g$. The main new ingredient is an integrated Ricci estimate for geodesics that minimize in moduli space and have thick endpoints, with no systole bound along their interiors.

arXiv comment27 pages. Comments welcome