Dominating surface-group representations via Fock-Goncharov coordinates

Let S be a punctured surface of negative Euler characteristic. We show that given a generic representation ρ:π₁(S) → PSLₙ(C), there exists a positive representation ρ₀:π₁(S) → PSLₙ(R) that dominates ρ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space Xₙ= PSLₙ(C)/PSU(n). Moreover, the ρ₀-lengths of peripheral curves remain unchanged. The dominating representation ρ₀ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.