Dominating surface-group representations into $\mathrm{PSL}_2 (\mathbb{C})$ in the relative representation variety

Abstract

Let $\rho$ be a representation of the fundamental group of a punctured surface into $\mathrm{PSL}_2 (\mathbb{C})$ that is not Fuchsian. We prove that there exists a Fuchsian representation that strictly dominates $\rho$ in the simple length spectrum, and preserves the boundary lengths. This extends a result of Gueritaud-Kassel-Wolff to the case of $\mathrm{PSL}_2 (\mathbb{C})$-representations. Our proof involves straightening the pleated plane in $\mathbb{H}^3$ determined by the Fock-Goncharov coordinates of a framed representation, and applying strip-deformations.