Cusps in Complex Boundaries of One-dimensional Teichmüller Space

This paper gives a proof of the conjectural phenomena on the complex boundary one-dimensional slices: Every rational boundary point is cusp shaped. This paper treats this problem for Bers slices, the Earle slices, and the Maskit slice. In proving this, we also obtain the following result: Every Teichmüller modular transformation acting on a Bers slice can be extended as a quasi-conformal mapping on its ambient space. We will observe some similarity phenomena on the boundary of Bers slices, and discuss on the dictionary between Kleinian groups and Rational maps concerning with these phenomena. We will also give a result related to the theory of L.Keen and C.Series of pleated varieties in quasifuchsian space of once punctured tori. Stony Brook IMS Preprint #2001/6 June 2001