The Moduli space of Riemann Surfaces of Large Genus

2012/02/26 by Alastair Fletcher, Fletcher, Alastair, Jeremy Kahn +3
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1202.5780

openalex publication_date 2012/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

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 (c₁g)^2g and bounded above by (c₂g)^2g, where the constants c₁,c₂ 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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