Dominating surface-group representations via Fock-Goncharov coordinates

Abstract

Let $S$ be a punctured surface of negative Euler characteristic. We show that given a generic representation $ρ:π_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{C})$, there exists a positive representation $ρ_0:π_1(S) \rightarrow \mathrm{PSL}_n(\mathbb{R})$ that dominates $ρ$ in the Hilbert length spectrum as well as in the translation length spectrum, for the translation length in the symmetric space $\mathbb{X}_n= \mathrm{PSL}_n(\mathbb{C})/\mathrm{PSU}(n)$. Moreover, the $ρ_0$-lengths of peripheral curves remain unchanged. The dominating representation $ρ_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.