# Coarse density of subsets of $M_g$

## TL;DR

This paper characterizes algebraic subvarieties of the moduli space of genus g Riemann surfaces that are coarsely dense with respect to Teichmüller or Thurston metrics, showing only the entire space can be dense.

## Contribution

It establishes a criterion for coarse density of algebraic subvarieties in the moduli space and applies it to orbit closures in the space of abelian differentials.

## Key findings

- Algebraic subvarieties are coarsely dense only if they are the entire moduli space.
- Identifies which strata of abelian differentials project densely to the moduli space.
- Provides a characterization of density for projections of GL_2(R)-orbit closures.

## 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 Teichm\"uller metric (or Thurston metric) if and only if it is all of $\mathcal{M}_g$. We apply this to projections of $\operatorname{GL}_2(\mathbb{R})$-orbit closures in the space of abelian differentials. Moreover, we determine which strata of abelian differentials have coarsely dense projection to $\mathcal{M}_g$.