turkmath.org

Much of two-dimensional hyperbolic geometry can be rephrased in terms of homomorphisms from the fundamental group of a surface to the Lie group SL_2(R). More precisely, in the space of all such homomorphisms, those that arise from hyperbolic geometry form a whole connected component called the Teichmüller component. Replacing SL_2(R) by SL_n(R), there similarly exists a preferred component in the space of all homomorphisms from a surface group to SL_n(R), called the Hitchin component. Hitchin proved that the Hitchin component is homeomorphic to an open ball of dimension 2(g-1)(n-1)(n+1) where g is the genus of the surface. I will present another parametrization of the Hitchin component, which is more closely connected to the geometric and dynamic properties of the homomorphisms involved. This is joint work with Guillaume Dreyer.