Remarks on the Hyperbolic Geometry of Product Teichmüller Spaces

Let e T be a cross product of n Teichmüller spaces of Fuchsian groups, n > 1. From the properties of Kobayashi metric and from the Royden-Gardiner theorem, e T is a complete hyperbolic manifold. Each two distinct points of e T can be joined by a hyperbolic geodesic segment, which is not in general unique. But when e T is finite dimensional or infinite dimensional of a certain kind, then among all such segments there is only one which enjoys a distinguished property: it is obtained from a uniquely determined holomorphic isometry of the unit disc into e T . If QC(G) is the Quasiconformal Deformation space of a finitely generated Kleinian group G, then since its holomorphic covering is a product of finite dimensional Teichmüller spaces, all the above results hold for QC(G).