The Moduli Space of Riemann Surfaces of Large Genus

Let Mg,ǫ be the ǫ-thick part of the moduli space Mg of closed genus g surfaces. In this article, we show that the number of balls of radius r needed to cover Mg,ǫ is bounded below by (c1g) 2g and bounded above by (c2g) , where the constants c1, c2 depend only on ǫ 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.