大亏格的Riemann Surfaces的模空间
The Moduli space of Riemann Surfaces of Large Genus
摘要：
设$\mathcal{M}_{g,ε}$是闭亏格$g$曲面的模空间$\mathcal{M}_g$的$ε$厚部分.在这篇文章中，我们证明了覆盖$\mathcal{M}_{g,ε}$所需的半径为$r$的球的数量由$(c_1g)^{2g}$限制在下面，由$(c_2g)^{2g}$限制在上面，其中常数$c_1,c_2$仅取决于$ε$和$r$，特别是不取决于$g$。利用计数结果，我们证明了存在任意大内射半径的黎曼曲面，它们不接近（在Teichmüller度量中）固定闭黎曼曲面的有限覆盖。这一结果说明了Escheriche preis猜想的尖锐性。
英文摘要：
Let $\mathcal{M}_{g,ε}$ be the $ε$-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 $\mathcal{M}_{g,ε}$ 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 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.
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