A divergent horocycle in the horofunction compactification of the Teichm\"uller metric
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.
0
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.