Fenchel-Nielsen coordinates for maximal representations

We develop Fenchel-Nielsen coordinates for representations of surface groups into with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichmuller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigation of the gluing. As applications we obtain results for non-closed surfaces, which have been known only for closed surfaces before: we count the number of connected components of their representation space and prove continuity for the limit curve for a certain type of representations.