On the properties of various compactifications of Teichmüller space

In this paper, we study the Gardiner–Masur compactification, the Teichmüller compactification and the asymptotic visual compactification of Teichmüller space. We prove that the Gardiner–Masur compactification is stronger than the Teichmüller compactification. We also prove that the asymptotic visual compactification is independent of the base point. Moreover, we prove that in these three compactifications, the convergences (of a sequence in Teichmüller space) to a same indecomposable measured foliation are equivalent. As an application, we construct some counter examples about the relation between the Gardiner–Masur compactification and the Thurston compactification, and the relation between the Teichmüller compactifications with different base points.