The nonabelian Hodge correspondence links the moduli space of representations of the fundamental group of a topological surface with a moduli space of holomorphic objects on Riemann surfaces known as Higgs bundles. After motivating these objects, I will describe how nilpotent elements of semisimple complex Lie algebras define interesting subvarieties of these moduli spaces called global Slodowy slices. The Slodowy slice construction generalizes the Higgs bundle description of a particularly interesting subset of representations known as the Hitchin representations. For some low rank examples, I will describe how every connected component of every Slodowy slice contains a generalization of Hitchin representations known as Anosov representations. In particular, this provides new examples of convex cocompact surface subgroups of SU(1,n) arising from variations of Hodge structure. The last part is joint work with Zach Virgilio.
