Geometry & Topology Seminar: On Line-Hyperplane Structures for Hitchin Representations

Abstract: Recall that Teichmüller space of a closed surface S (of genus at least two) can be viewed as the moduli space of (marked) hyperbolic structures on S. For the group G = SL(n,R), or more generally G a split, real, simple Lie group, we shall define a special connected component of representations of the fundamental group of S into G, the G-Hitchin-component, that can be similarly ‘geometrized.’ By this we mean the following: there is a mysterious higher dimensional manifold M, fibering over the surface S, that carries locally homogeneous (G,X)-structures that determine, and are determined by, G-Hitchin representations.

                There are many natural questions to ask about this remarkable construction. An obvious one is as follows: what is the fiber F of M over S? After appropriate background, we shall explore this question in the case of G = SL(n,R) and X is a certain flag manifold of SL(n,R), namely the space of line-hyperplane pairs in R^n. The talk features a mixture of concrete geometry as well as an application of Wall’s classification of highly connected odd-dimensional manifolds. This project is joint work with Andrea Tamburelli.

Host: Parker Evans