Convergence of earthquake and horocycle paths to the boundary of Teichm\"uller space

TL;DR

This paper investigates how earthquake and horocycle paths in Teichmüller space converge to the boundary, revealing new convergence properties related to measured geodesic laminations and quadratic differentials.

Contribution

It establishes convergence of earthquake and horocycle paths to the Gardiner-Masur boundary under specific ergodic conditions, connecting different boundary theories in Teichmüller space.

Findings

Earthquake paths directed by uniquely ergodic laminations converge to the boundary.

Horocycle paths induced by quadratic differentials with unique ergodic vertical foliations converge to the boundary.

The embedding of flat metrics into geodesic currents aids in proving boundary convergence.

Abstract

We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichm\"uller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into the space of geodesic currents, we prove that a horocycle path in Teichm\"uller space, induced by a quadratic differential whose vertical measured foliation is unique ergodic, converges to the Gardiner-Masur boundary and to the Thurston boundary.