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 a...
