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.