Coarse density of subsets of moduli space

Dozier, Benjamin; Sapir, Jenya

Annales de l'Institut Fourier, Tome 71 (2021) no. 3, pp. 1121-1134.

Nous montrons qu’une sous-variété algébrique de l’espace de module des courbes de genre $g$ est grossièrement dense pour la métrique de Teichmuller (ou la métrique de Thurston) si et seulement si la variété est de dimension maximale. En guise d’application, nous déterminons les strates des différentielles abéliennes dont la projection est grossièrement dense dans l’espace de modules. De plus, nous obtenons un résultat sur les clôtures des $G{L}_2\left(ℝ\right)$ orbites dans l’espace des différentielles abéliennes pour lequel les projections sont grossièrement denses.

We show that an algebraic subvariety of the moduli space of genus $g$ Riemann surfaces is coarsely dense with respect to the Teichmüller metric (or Thurston metric) if and only if it has full dimension. We apply this to determine which strata of abelian differentials have coarsely dense projection to moduli space. Furthermore, we prove a result on coarse density of projections of $G{L}_2\left(ℝ\right)$-orbit closures in the space of abelian differentials.

Classification : 57M50, 30F60, 32G15
Keywords: Teichmüller Theory, Subvarieties of moduli space, Abelian differentials
Mot clés : Teichmüller Theory, Subvarieties of moduli space, Abelian differentials

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