Quasi-isometric embeddings for shrinking maps from surfaces into the moduli space

TL;DR

This paper studies when certain maps from hyperbolic surfaces into the moduli space are quasi-isometric embeddings, showing that under mild conditions, their properties depend only on their monodromy, with applications to holomorphic maps.

Contribution

It provides a characterization of quasi-isometric embeddings for shrinking maps into the moduli space based solely on monodromy, under mild conditions.

Findings

Characterization of quasi-isometric embeddings via monodromy

Application to holomorphic maps

Conditions under which the properties hold

Abstract

We investigate shrinking maps from a cusped hyperbolic surface into the moduli space of closed Riemann surfaces. For such a map and its lift to the Teichm\"uller space, we consider whether they are quasi-isometric embeddings with respect to natural metrics like the Teichm\"uller distance and the intrinsic distance. Under a mild condition, we prove that these properties are characterised solely by the map's monodromy. These characterisations apply, in particular, to holomorphic maps.