Explicit upper bound for the Weil-Petersson volumes

TL;DR

This paper derives an explicit upper bound for Weil-Petersson volumes of moduli spaces of punctured Riemann surfaces, demonstrating growth rate bounds as genus increases, using Penner's combinatorial approach.

Contribution

It provides the first explicit constant-based upper bound for Weil-Petersson volumes, applicable for fixed puncture counts and large genus, using combinatorial integration methods.

Findings

Weil-Petersson volume grows at most as c^g g^{2g} for fixed punctures as genus increases.

Explicit constant c is derived and independent of the number of punctures.

The bound is obtained via Penner's combinatorial integration scheme with embedded trivalent graphs.

Abstract

An explicit upper bound for the Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces is obtained, using Penner's combinatorial integration scheme with embedded trivalent graphs. It is shown that for a fixed number of punctures n and for genus g going to infinity, the Weil-Petersson volume of M_{g,n} has an upper bound c^g g^{2g}. Here c is an independent of n constant, which is given explicitly.