Let $S_{g,k}$ be a connected oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$ be two representations. $\rho_2$ is said to dominate $\rho_1$ if there exists $\lambda \le 1$ such that $\ell_{\rho_1}(\gamma) \le \lambda \cdot \ell_{\rho_2}(\gamma)$ for all $\gamma \in \pi_1(S_{g,k})$, where $\ell_{\rho}(\gamma)$ denotes the translation length of $\rho(\gamma)$ in $\mathbb{H}^3$. In 2016, Deroin–Tholozan showed that for a closed surface $S$ and a non-Fuchsian representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S_{g,k}) \rightarrow PSL_2(\mathbb{R})$ that strictly dominates $\rho$. In 2023, Gupta–Su proved a similar result for punctured surfaces, where the representations lie in the same relative representation variety. Here, we generalize these results to the case of higher-rank representations.

For a representation $\rho : \pi_1(S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, the Hilbert length of a curve $\gamma\in \pi_1(S_{g,k})$ for $n >2$ is defined as \begin{equation} \ell_{\rho}(\gamma):=\ln \Bigg| \frac{\lambda_n}{\lambda_1} \Bigg|, \end{equation} where $\lambda_n$ and $\lambda_1$ are the largest and smallest eigenvalues of $\rho(\gamma)$ in modulus respectively. We show that for any generic representation $\rho : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S_{g,k}) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock-Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$ representation. Weighted planar networks and the Collatz–Wielandt formula for totally positive matrices play a crucial role.

Let $X_n$ be the symmetric space of $PSL_n(\mathbb{C})$. The translation length of $A\in PSL_n(\mathbb{C})$ in $X_n$ is given as \begin{equation} \tau(A)= \sum_{i=1}^{n}\log |\lambda_i(A)|^2, \end{equation} where $\lambda_i(A)$ are the eigenvalues of $A$. We show that the same $j$ dominates $\rho$ with respect to the translation length at the origin as well. Lindström’s Lemma for planar networks and Weyl’s Majorant Theorem are some of the key ingredients of the proof.

In both cases, if $S_{g,k}$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.