On Thurston's Stretch Lines in Teichm\"uller Space

TL;DR

This paper investigates the asymptotic behavior of measured geodesic lamination lengths along Thurston's stretch lines in Teichmüller space, providing insights into the geometric structure of these geodesics under Thurston's asymmetric metric.

Contribution

It offers a detailed analysis of the asymptotic length behavior of laminations along stretch lines, advancing understanding of Thurston's metric geometry.

Findings

Lengths of measured laminations grow or shrink asymptotically along stretch lines.

Provides explicit asymptotic formulas for lamination lengths.

Enhances understanding of geodesic behavior in Thurston's metric space.

Abstract

The Teichm\"uller space $\mathcal{T}(\Sigma)$ of a surface $\Sigma$ is equipped with Thurston's asymmetric metric. Stretch lines are oriented geodesics for this metric on $\mathcal{T}(\Sigma)$. We give the asymptotic behavior of the lengths of the measured geodesic laminations as one follows a stretch line in the positive direction.