The Moduli space of Riemann Surfaces of Large Genus

TL;DR

This paper investigates the geometric complexity of the moduli space of large genus Riemann surfaces, providing bounds on covering numbers and demonstrating the existence of surfaces with large injectivity radius that challenge certain conjectures.

Contribution

It establishes bounds on the covering numbers of the thick part of the moduli space and shows the existence of large injectivity radius surfaces not close to finite covers, highlighting the sharpness of the Ehrenpreis conjecture.

Findings

Bounding the number of balls needed to cover the moduli space by functions of genus

Existence of Riemann surfaces with arbitrarily large injectivity radius not close to finite covers

Results support the sharpness of the Ehrenpreis conjecture

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 $(c_1g)^{2g}$ and bounded above by $(c_2g)^{2g}$, 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 Teichm\"uller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates the sharpness of the Ehrenpreis conjecture.