The Moduli Space of Riemann Surfaces of Large Genus

Let M_(g,ϵ) be the ϵ -thick part of the moduli space M_g of closed genus g surfaces. In this article, we show that the number of balls of radius r needed to cover M_(g,ϵ) is bounded below by (c_1g)^(2g) and bounded above by (c_2g)^(2g), where the constants c_1, c_2 depend only on ϵ and r, and in particular not on g. Using this counting result we prove that there are Riemann surfaces of arbitrarily large injectivity radius that are not close (in the Teichmüller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates the sharpness of the Ehrenpreis conjecture