Schottky groups and Bers boundary of Teichmüller space (Q1411274)

In this paper the action of the mapping class group on a Bers slice is extended to a wider class of Kleinian groups. Let \(S\) be an oriented compact surface possibly with boundary, and let \(T(S)\) be the Teichmüller space. Let \(R(S)\) be the space of conjugacy classes \([\rho ,G]\) of representations \(\rho :\pi _{1}(S)\rightarrow G\subset \) PSL\(_{2}(\mathbb{C}) \) of \(\pi _{1}(S)\) which map each component of the boundary \(\partial (S)\) to parabolic elements. Let \(X\in T(S)\) and denote by \(B_{X}\) and \(C_{X}\) the Bers slice and the extended Bers slice, respectively. A representation \( [\rho ,G]\) is an element of \(C_{X}\) if \(G\) is a function group with an invariant component \(\Omega _{0}(G)\) of the region of discontinuity \(\Omega (G)\) which is covering \(X\). We have that the closure \(\overline{B}_{X}\) is contained in \(C_{X}\). The Bers boundary is \(\partial B_{X}=\overline{B} _{X}-B_{X}\). Let \(S\) be a closed surface of genus \(\geq 2\) and let \(S_{X}\) be the subset of \(C_{X}\) consisting of Schottky groups. The author obtains a sufficient condition for the action of the mapping class group to be continuous at a given point in \(C_{X}\): Let \([\rho ,G]\) be an element of \(C_{X}\) such that all components of \(\Omega (G)/G\) except for \(X=\Omega _{0}(G)/G\) have no moduli of deformation then Mod(\(S\)) acts continuously at \([\rho ,G]\). It is also proved that the orbit under the action of Mod(\(S\)) of every maximal cusp is dense in the Bers boundary. Combining both results the following Theorem is obtained: Let \(S\) be a closed surface of genus \(\geq 2\). The set of accumulation points of \(S_{X}\) contains \(\partial B_{X}\). This result was proved in [\textit{J. P. Otal}, C. R. Acad. Sci., Paris, Sér. I 316, No. 2, 157--160 (1993; Zbl 0772.30039)] but here a different proof is given. The Thurston's compactness theorem plays an important role. The author also collects some properties of \(S_{X}\) in several Lemmas at the end of the paper.