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

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.

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character variety, surface group, mapping class group
Mathematical Subject Classification
Primary: 57M50
Secondary: 57M05, 20E36, 20F29
Received: 17 April 2020
Revised: 3 March 2021
Accepted: 10 April 2021
Published: 15 June 2022
Proposed: David M Fisher
Seconded: Ulrike Tillmann, Benson Farb
Indranil Biswas
School of Mathematics
Tata Institute of Fundamental Research
Subhojoy Gupta
Department of Mathematics
Indian Institute of Science
Mahan Mj
School of Mathematics
Tata Institute of Fundamental Research
Junho Peter Whang
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States