The curve complex and covers via hyperbolic 3-manifolds

TL;DR

This paper provides a new proof that the relation between curve complexes induced by surface coverings is a quasi-isometric embedding, utilizing hyperbolic 3-manifolds for distance estimation.

Contribution

It introduces an alternative proof of a known result by employing hyperbolic 3-manifold techniques for distance estimation.

Findings

Curve complexes relation is a quasi-isometric embedding

Hyperbolic 3-manifolds can be used for distance estimates

New proof offers different perspective on the result

Abstract

Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via hyperbolic 3-manifolds.