GROWTH OF THE WEIL-PETERSSON INRADIUS OF MODULI SPACE

In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichmfiller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus g with n punctures as a function of g and n. We show that the Weil-Petersson inradius is comparable to root ln g with respect to g, and is comparable to 1 with respect to n. Moreover, we also study the asymptotic behavior, as g goes to infinity, of the Weil-Petersson volumes of geodesic balls of finite radii in Teichmfiller space. We show that they behave like o(( 1/g)((3-epsilon)g)) as g -> infinity, where epsilon > 0 is arbitrary.