Parametrizing Hitchin components

We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller 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 λin S with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of λ; and triangle invariants associated with each component of the complement S-λ. We describe identities and relations satisfied by these invariants, and use the resulting coordinates to parametrize the Hitchin component.

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