Canonical representations of surface groups

Abstract

Let $\Sigma_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group $\mathrm{Mod}_{g,n}$ of $\Sigma_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if \[\rho: \pi_1(\Sigma_{g,n})\to \mathrm{GL}_r(\mathbb{C})\]is a representation whose conjugacy class has finite orbit under $\mathrm{Mod}_{g,n}$, and $r \lt\sqrt{g+1}$, then $\rho$ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson's integrality conjecture for cohomologically rigid local systems.