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.

15 pages, 3 figures, v2 corrects our handling of the degenerate and co-axial case