Geometric Structures and Representations of Surface Groups

Representations of hyperbolic groups into higher rank Lie groups has been an active topic of study in recent years. In particular the character variety associated with a surface group for some semi-simple Lie group of non-compact type admits remarkable connected components containing only discrete and faithful representations. A union of such connected components is called a higher rank Teichmüller space. In all the known cases, the representations in these components all satisfy an Anosov property, which is a dynamical property stronger than being discrete and faithful. Some of these spaces can be interpreted as spaces of geometric structures: as for instance convex projective structures on surfaces, or fibered photon structures. 

In this thesis, we bring original contributions to this area, focusing in particular on the locally symmetric space and parabolic structures associated to Anosov representations. The first part of this thesis is rather general, and discuss parabolic structures constructed using a domain of discontinuity as well as their relation with the locally symmetric space for certain Anosov representations.