Canonical representations of surface groups

Let Σg,n be an orientable surface of genus g with n punctures. We study actions of the mapping class group Modg,n of Σg,n via Hodge-theoretic and arithmetic techniques. We show that if \[ ρ: π_1(Σg,n)→ GL_r(C) \] is a representation whose conjugacy class has finite orbit under Modg,n, and r <√g+1, then ρ 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.