Let M be the moduli space of surface group representations in a real reductive Lie group G. The most basic topological question one can ask about M is the determination of its connected components. For example, when G=PSL(2,R), Goldman proved that M has 4g-3 components, two of which can be identified with Teichmüller space. Much progress has been made on this problem over the last 30-40 years but it is, in general, still open. A conjectural answer to the question - based on Hitchin's approach via non-abelian Hodge Theory - can be stated in terms of the generalised Cayley correspondence for moduli spaces of G-Higgs bundles, and is closely related to higher Teichmüller theory.
The talk is mainly based on joint work with Steve Bradlow, Brian Collier, Oscar Garcia-Prada, and André Oliveira.