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Given an oriented surface of positive genus with finitely many punctures, weclassify the finite orbits of the mapping class group action on the modulispace of semisimple complex special linear two dimensional representations ofthe fundamental group of the surface. For surfaces of genus at least two, suchorbits correspond to homomorphisms with finite image. For genus one, theycorrespond to the finite or special dihedral representations. We also obtain ananalogous result for bounded orbits in the moduli space.

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