Canonical representations of surface groups

Let ₆, ₍ be an orientable surface of genus g with n punctures. We study actions of the mapping class group Mod₆, ₍ of ₆, ₍ via Hodge-theoretic and arithmetic techniques. We show that if \: ₁ (₆, ₍) GLᵣ (C) \ is a representation whose conjugacy class has finite orbit under Mod₆, ₍, and r +1, then 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.