The Moduli space of Riemann Surfaces of Large Genus

Abstract

Let ${\mathcal{M}}_{g,\epsilon }$ be the $\epsilon$-thick part of the moduli space ${\mathcal{M}}_g$ of closed genus $g$ surfaces. In this article, we show that the number of balls of radius $r$ needed to cover ${\mathcal{M}}_{g,\epsilon }$ is bounded below by $(c_{1}g{)}^{2g}$ and bounded above by $(c_{2}g{)}^{2g}$, where the constants $c_1,c_2$ depend only on $\epsilon$ and $r$, and in particular not on $g$. Using the 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.

v2, accepted in GAFA, updates based on referee's comments

References in corpus (1)

• Counting Essential Surfaces in a Closed Hyperbolic 3-Manifold