A divergent horocycle in the horofunction compactification of the Teichm\"uller metric

TL;DR

This paper presents an example of a horocycle in Teichmüller space that diverges in the horofunction compactification, revealing new insights into the boundary behavior of Teichmüller metrics.

Contribution

It provides the first explicit example of a divergent horocycle in the Teichmüller space of a punctured sphere, extending to higher dimensions through covering constructions.

Findings

Horocycle in Teichmüller space does not converge in the compactification.

Existence of a simple closed curve with periodic but non-constant extremal length.

The example applies to all higher-dimensional Teichmüller spaces via coverings.

Abstract

We give an example of a horocycle in the Teichm\"uller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichm\"uller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is periodic but not constant along the horocycle. The example lifts to any Teichm\"uller space of complex dimension greater than one via covering constructions.