Hyperbolic spaces in Teichm\"uller spaces

TL;DR

This paper demonstrates that hyperbolic spaces of any dimension can be embedded into Teichmüller spaces of surfaces, revealing new geometric relationships and implications for the structure of curve complexes.

Contribution

It proves the existence of almost-isometric embeddings of hyperbolic spaces into Teichmüller spaces for any dimension, with the embeddings lying in the thick part.

Findings

H^n embeds into Teichmüller space with quasi-isometry.

H^n quasi-isometrically embeds into the curve complex.

Embeddings are contained in the thick part of Teichmüller space.

Abstract

We prove, for any n, that there is a closed connected orientable surface S so that the hyperbolic space H^n almost-isometrically embeds into the Teichm\"uller space of S, with quasi-convex image lying in the thick part. As a consequence, H^n quasi-isometrically embeds in the curve complex of S.