PARAMETERIZING HITCHIN COMPONENTS

We construct a geometric, real-analytic parameterization of the Hitchin component Hit(n)(S) of the PSLn(R)-character variety R-PSLn (R)(S) of a closed surface S. The approach is explicit and constructive. In essence, our parameterization is an extension of Thurston 's shearing coordinates for the Teichmiiller space of a closed surface, combined with Fock-Goncharov's coordinates for the moduli space of positive framed local systems of a punctured surface. More precisely, given a maximal geodesic lamination lambda subset of S with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of lambda and triangle invariants associated with each component of the complement S - lambda. We describe identities and relations satisfied by these invariants, and we use the resulting coordinates to parameterize the Hitchin component.