On the non-Archimedean Hitchin map for ${\mathrm{S}\mathrm{L}}_2(F)$

Let $F$ be a non-Archimedean valued field, $\mathrm{\Sigma }$ a closed Riemann surface of genus at least two, and $\mathrm{\Gamma }$ its fundamental group. Building on the theory of equivariant harmonic maps into $\mathbb{R}$-trees, we study the non-Archimedean Hitchin map from the ${\mathrm{S}\mathrm{L}}_2(F)$-character variety ${\mathcal{X}}_F(\mathrm{\Gamma })$, equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $\mathrm{\Sigma }$. We prove that this map is continuous and that its image is contained in the space of Jenkins–Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of $\mathrm{\Gamma }$ on the Bruhat–Tits tree of ${\mathrm{S}\mathrm{L}}_2(F)$ is never small.