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 $\ddot{I}:\ddot{I}{€}_1(S)\to {\mathrm{PSL}}_n(\mathbb{C})$, there exists a positive representation $\ddot{I}{}_0:\ddot{I}{€}_1(S)\to {\mathrm{PSL}}_n(\mathbb{R})$ that dominates $\ddot{I}$ 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 $\ddot{I}{}_0$-lengths of peripheral curves remain unchanged. The dominating representation $\ddot{I}{}_0$ is explicitly described via Fock-Goncharov coordinates. Our methods are linear-algebraic, and involve weight matrices of weighted planar networks.

41 pages, v2 incorporates several minor improvements, to appear in Geometriae Dedicata