turkmath.org

Representations of the fundamental group of an orientable surface of finite type into the real symplectic group Sp(2n,R) with maximal Toledo invariant are of particular interest in Higher Teichmüller theory. These representations generalize Fuchsian representations of the fundamental group of a surface into PSL(2,R). Moreover, maximal representations have particularly nice properties, e.g. they are injective with a discrete image in G. In my talk, I introduce coordinates on the space of maximal representations of the fundamental group of a punctured surface into Sp(2n,R). These coordinates generalize the Thurston's shear coordinates on the Teichmüller space. If time permits, I will explain how we can understand the topology of the space of maximal representations using these coordinates. This is a joint work with D. Alessandrini, O. Guichard and A. Wienhard.