Holomorphic curves in moduli spaces are quasi-isometrically immersed

Abstract

A holomorphic curve in moduli spaces is the image of a non-constant holomorphic map from a cusped hyperbolic surface $B$ of type $(g,n)$ to the moduli space ${\mathcal{M}}_h$ of closed Riemann surfaces of genus $h$. We explore the relationship between the holomorphicity and the Teichmüller distance. Our result shows that, when all peripheral monodromies are of infinite order, the holomorphic map is a quasi-isometric immersion, with parameters depending only on $g$, $n$, $h$ and the systole of $B$. Moreover, under an additional condition on the peripheral monodromies, the lifting embeds a fundamental domain of the hyperbolic surface $B$ into the Teichmüller space as a quasi-isometric embedding.