Anosov Flows, Surface Groups and Curves in Projective Space

Abstract

In 1990, Hitchin's proved a component of the space of representations of a surface group in SL(n,R) is homeomorphic to a ball. For n=2,3 this component has been identified with the holonomies of geometric structures (hyperbolic for n=2, or real projective for n=3). In the preprint "Anosov flows, Surface groups and Curves in Projective Space", we extend this interpretation to higher dimension and show every representation in Hitchin's component is attached to a (special) curve in projective space, thus giving a geometric interpretation of these representations. We also prove these representations are faithful, discrete and purely loxodromic (or hyperbolic)