Vertical Limits of Graph Domains
  We consider the limiting behavior of Teichm\"uller geodesics in the universal Teichm\"uller space $T(\mathbb{H})$. Our main result states that the limits of the Teichm\"uller geodesics in the Thurston's boundary of $T(\mathbb{H})$ may depend on both vertical and horizontal foliation of the corresponding holomorphic quadratic differential. 
Hrant Hakobyan and Dragomir Saric
math.CV