Rational approximation for Hitchin representations

TL;DR

The paper proves that Hitchin representations with image in $\mathrm{SL}(n,\mathbf{Q})$ are dense in the Hitchin component $\mathcal{H}_n(S)$ for genus $g\ge 3$, providing a dynamical proof using generalized twist flows $\Xi_\gamma^t$. It further shows that the same mechanism yields density for $\mathcal{H}_{\mathrm{Sp}(2k)}(S)$ and $\mathcal{H}_{G_2}(S)$ when restricted to $\mathbf{Q}$-points, by leveraging centralizer density and rational embeddings from $\mathrm{SL}(2)$, with the Platonov–Rapinchuk density theorem as a key arithmetic tool. The approach generalizes to other split groups, offering a framework to study arithmetic properties of surface subgroups in higher rank and guiding questions about extending to broader $\mathbf{Q}$-groups. Together, these results illuminate the $\mathbf{Q}$-rational structure of Hitchin components and their arithmetic consequences in lattice and building-theoretic contexts.

Abstract

A consequence of Rapinchuk et al. is that for $S$ a closed surface of genus $g\geq 2$, the set of Hitchin representations of $π_1(S)$ with image in $\mathrm{SL}(n,\mathbb{Q})$ is dense in the Hitchin component. We give a dynamical proof of this fact provided that $g\geq 3$. Moreover, we extend it to some other $\mathbb{Q}$-groups such as $\mathrm{Sp}(2k,\mathbb{Q})$ and $\mathrm{G}_2(\mathbb{Q})$, where the results are new.

Theorems & Definitions (15)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Lemma 2.2
• Remark
• Theorem 2.3: Goldman86*Theorem 4.7
• Theorem 2.4: BCL*Proposition 10.1
• proof : Proof of Lemma \ref{['bendconnection']}
• Theorem 3.1: platonovrapinchuk*Theorem 7.7
• Corollary 3.2