The asymptotic geometry of the Teichm\"uller metric

TL;DR

This paper investigates the asymptotic behavior of extremal length along Teichmüller rays, characterizes boundary points in the Gardiner-Masur boundary, and explores the structure of Busemann points and the detour metric.

Contribution

It provides a detailed analysis of the asymptotic geometry of the Teichmüller metric, including boundary descriptions and convergence criteria for Busemann points.

Findings

Calculated endpoints of Teichmüller rays in the boundary.

Proved the equivalence of Gardiner-Masur and horofunction compactifications.

Characterized Busemann points and determined the detour metric.

Abstract

We determine the asymptotic behaviour of extremal length along arbitrary Teichm\"uller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichm\"uller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the set of Busemann points. We show that the Busemann points are exactly the limits of the Teichm\"uller rays, and we give a necessary and sufficient condition for a sequence of Busemann points to converge to a Busemann point. Finally, we determine the detour metric on the boundary.