Vertical Limits of Graph Domains

Abstract

We consider the limiting behavior of Teichmüller geodesics in the universal Teichmüller space $T(\mathbb{H})$. Our main result states that the limits of the Teichmüller geodesics in the Thurston's boundary of $T(\mathbb{H})$ may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential.