#### Volume 26, issue 2 (2022)

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Surface group representations in ${\rm SL}_2(\mathbb{C})$ with finite mapping class orbits

### Indranil Biswas, Subhojoy Gupta, Mahan Mj and Junho Peter Whang

Geometry & Topology 26 (2022) 679–719
##### 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.

##### Keywords
character variety, surface group, mapping class group
##### Mathematical Subject Classification
Primary: 57M50
Secondary: 57M05, 20E36, 20F29