Notes on projective structures and Kleinian groups
Let \(G\) be a Fuchsian group, \(D\) the unit disk in the complex plane \(\mathbb{C}\), \(\widehat C = \mathbb{C} \cup \{\infty\}\), and \(R\) the hyperbolic orbifold \(D/G\). \(PSL(2,\mathbb{C})\) denotes the group of Möbius transformations of the sphere \(\widehat C\). A projective structure on \(R\) is defined by a pair consisting of a representation \(M:G \to PSL (2,C)\) and a locally univalent holomorphic \(M\)-equivariant map \(f:D \to \widehat C\). One defines an equivalence relation on the set of projectives structures. The space of equivalent classes is, via a one-to-one map, the affine space of holomorphic quadratic differentials on \(R\). The space of norm-bounded quadratic differentials on \(R\), and the corresponding projective structures are called bounded projective structures. In the paper under review the following results are established: 1. Bounded covering projective structures with distinct kernels are in distinct components of the set of bounded covering projective structures. 2. If \(R\) is a hyperbolic Riemann surface of finite area, and \(Q\) a bounded covering projective structure associated to \(R\), then if \(\text{Ker} (M_ Q)\) is such that \(D/ \text{Ker} (M_ Q)\) is conformally a plane domain, then \(f_ Q(D)\) is conformally equivalent to \(D/ \text{ker} (M_ Q)\). The boundedness of the covering projective structures is also approached, and the structure of the interior of the set of bounded discrete projective structures is studied.
scientific article; zbMATH DE number 639564