Let Σ g,n be an orientable surface of genus g with n punctures. We study actions of the mapping class group Mod g,n of Σ g,n via Hodge-theoretic and arithmetic techniques. We show that if ρ : π 1 (Σ g,n ) → GL r (C) is a representation whose conjugacy class has finite orbit under Mod g,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. CONTENTS 1. Introduction 1 2. Representation-theoretic preliminaries 10 3. Hodge-theoretic preliminaries 20 4. The period map associated to a unitary representation 23 5. The main cohomological results 27 6. The asymptotic Putman-Wieland conjecture 30 7. Proof of the main theorem on MCG-finite representations 33 8. Consequences for arithmetic representations 39 9. Questions and examples 43