Let $S$ be an oriented surface of negative Euler characteristic and $\rho_1,\ \rho_2:\pi_1(S) \rightarrow PSL_2(\mathbb{C})$ 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 $\gamma \in \pi_1(S)$, 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 $\rho : \pi_1(S) \rightarrow PSL_2(\mathbb{C})$, there exists a Fuchsian representation $j : \pi_1(S) \rightarrow PSL_2(\mathbb{R})$ 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 $\rho : \pi_1(S) \rightarrow PSL_n(\mathbb{C})$ where $n >2$, the Hilbert length of a curve $\gamma\in \pi_1(S)$ 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 $\rho : \pi_1 (S) \rightarrow PSL_n(\mathbb{C})$, there is a Hitchin representation $j : \pi_1 (S) \rightarrow PSL_n(\mathbb{R})$ that dominates $\rho$ in the Hilbert length spectrum. The proof uses Fock–Goncharov coordinates on the moduli space of framed $PSL_n(\mathbb{C})$-representations. 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} \ell_{X_n}(A)= \sum_{i=1}^{n}\log (\sigma_i(A))^2, \end{equation} where $\sigma_i(A)$ are the singular values of $A$. We show that the same $j$ dominates $\rho$ in the translation length spectrum as well. Lindström’s Lemma for planar networks is one of the key ingredients of the proof.

In both cases, if $S$ is a punctured surface, then $j$ lies in the same relative representation variety as $\rho$.

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Last updated: 02 Nov 2024