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

TL;DR

This paper investigates the non-Archimedean Hitchin map for $ ext{SL}_2(F)$, demonstrating its continuity, the containment of its image within Jenkins--Strebel differentials, and characterizing unbounded representations via group actions on Bruhat--Tits trees.

Contribution

It establishes the continuity of the non-Archimedean Hitchin map and provides a dynamical criterion for unbounded representations in the non-Archimedean setting.

Findings

The Hitchin map is continuous.

Its image lies in Jenkins--Strebel differentials.

Unbounded representations induce non-small actions on Bruhat--Tits trees.

Abstract

Let $F$ be a non-Archimedean valued field, $\Sigma$ a closed Riemann surface of genus at least two, and $\Gamma$ its fundamental group. Building on the theory of equivariant harmonic maps into $\mathbb{R}$-trees, we study the non-Archimedean Hitchin map from the $\mathrm{SL}_2(F)$-character variety $\mathcal{X}_F(\Gamma)$, equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $\Sigma$. 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 $\Gamma$ on the Bruhat--Tits tree of $\mathrm{SL}_2(F)$ is never small.