An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces

Abstract.

An explicit upper bound for the Weil-Petersson volumes of punctured Riemann surfaces is obtained using Penner's combinatorial integration scheme from [4]. It is shown that for a fixed number of punctures n and for genus g increasing,

$\lim\limits_{g\to\infty, n{\rm fixed}}\frac{\ln{\rm vol}_{WP} (\M_{g,n})}{g\ln g}\le 2,$

while this limit is exactly equal to two for n=1.