Surface group representations in SL2(ℂ) with finite mapping class orbits

Abstract

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space. 57M50; 57M05, 20E36, 20F29 1. Introduction 679 2. Background 683 3. Non-Zariski-dense representations 690 4. Analysis of Dehn twists 696 5. Proof of the main results 701 6. Applications 712 Appendix Case of a once-punctured torus 713 References 717