The relationship between the compactifications of by Bers and Thurston was studied by Kerckhoff and Thurston in . To explain the results of that paper we need to introduce a little notation. Let denote the Teichmüller space of a compact surface of genus . Given a pair of points and in , there is a quasi-Fuchsian group , with two invariant components, and , such that and . Conversely, a pair in determines a quasi-Fuchsian group with an isomorphism to the fundamental group of . Fixing a element in the first factor of we get an embedding of Teichmüller space into the set of conjugacy classes of representations of into . The space has a topology of convergence on finite system of generators. This embedding is holomorphic, and it is equivalent to the Bers embedding; the image is called a Bers slice, and its compactification is the Bers compactification of Teichmüller space. Changes of base point (the fixed point in the first factor of ) induce biholomorphic mappings between Bers slices.
It is also shown in the same paper that Bers' and Thurston's compactifications of Teichmüller space are not equivalent; this is a consequence of the following theorem.
Pablo Ares Gastesi 2005-08-31