Surface Group Representations

To go beyond discrete, faithful representations of surface groups into PSL(2, R) one starts with the canonical irreducible representation of PSL(2, R) into PSL(n, R). Next, with a Fuchsian representation as a starting point, one deforms such a representation in PSL(n,R). The set of such representations is called the Hitchin component. In the early 90s, Hitchin proved that the set of these representations forms a connected component of the character variety, and is homeomorphic to a ball. The internal geometry of such representations became clearer in around 2006.

Labourie and Fock-Goncharov gave two different ways of understanding the Hitchin component in 2006. Labourie’s approach was dynamical in nature, while Fock-Goncharov’s was algebro-geometric. The first approach gave rise to Anosov representations generalizing Hitchin representations in one direction. The second approach gave rise to positive representations generalizing Hitchin representations in another direction. This has led to a flurry of activity over the last two decades. A recurring theme in the field is to connect such representations with geometric (G,X) structures in the sense of Klein and Thurston, notably when X is a generalized flag space associated to a semi-simple G.

The field is around two decades old and is now approaching maturity, with foundations now laid and a large number of open avenues of inquiry.