Coarse density of subsets of $M_g$

Abstract

Let ${\mathcal{M}}_g$ be the moduli space of genus $g$ Riemann surfaces. We show that an algebraic subvariety of ${\mathcal{M}}_g$ is coarsely dense with respect to the Teichmüller metric (or Thurston metric) if and only if it is all of ${\mathcal{M}}_g$. We apply this to projections of ${\mathrm{GL}}_2(\mathbb{R})$-orbit closures in the space of abelian differentials. Moreover, we determine which strata of abelian differentials have coarsely dense projection to ${\mathcal{M}}_g$.

12 pages, 1 figure

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