Rational approximation for Hitchin representations

Abstract

A consequence of Rapinchuk et al. is that for $S$ a closed surface of genus $g\ge 2$, the set of Hitchin representations of $\ddot{I}{€}_1(S)$ with image in $\mathrm{SL}(n,\mathbb{Q})$ is dense in the Hitchin component. We give a dynamical proof of this fact provided that $g\ge 3$. 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.

9 pages, to appear in Groups, Geometry and Dynamics