On Arc Coordinates for Maximal Representations

Given a hyperbolic surface with boundary, arc coordinates provide a parametrization of the Teichmüller space. They rely on the choice of a family of arcs which start and end at boundary components and are orthogonal to them. Higher rank Teichmüller theories are a generalization of classical Teichmüller theory and are concerned with the study of representations of the fundamental group of an oriented surface of negative Euler characteristic into simple real Lie groups G of higher rank. It is well known that maximal representations are a higher rank Teichmüller theory for G Hermitian. In this thesis we will discuss how to generalize arc coordinates for maximal representations, focusing on the case where the surface is a pair of pants and G is the symplectic group PSp(4,R). This will be possible by introducing geometric parameters on the space of right-angled hexagons in the Siegel space X, which lead to the visualization of a right-angled hexagon as a polygonal chain inside the hyperbolic plane. We discuss geometric properties of reflections in X and introduce the notion of maximal representations of a reflection group. We give a parametrization of maximal representations of the reflection group into PSp(4,R), which allows us to parametrize a subset of maximal and Shilov hyperbolic representations into PSp(4,R).