On Completing Triangles in Teichmüller Space

Guowu Yao

Let $T(\Delta)$ be the universal Teichmüller space. Three points $[f]$, $[g]$ and $[h]$ in $T(\Delta)$ are called to form a completing triangle if each pair of them has a unique geodesic joining them. Recently, Z. Zhou and L. Liu constructed two Strebel points $[f]$ and $[g]$ such that $[id]$, $[f]$ and $[g]$ form a non-completing triangle. The computation in their construction is lengthy and complicated. In this note, it is shown that their results can be obtained in much simpler a way. Indeed, the current theory of Teichmüller spaces allows us to give more information on triangles in an infinite-dimensional Teichmúller space. Our method is self-contained and applies for general Teichmüller spaces.