New York Journal of Mathematics
Volume 24 (2018) 241-250

  

Indranil Biswas, Thomas Koberda, Mahan Mj, and Ramanujan Santharoubane

Representations of surface groups with finite mapping class group orbits

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Published: February 28, 2018
Keywords: Representation variety; surface group; mapping class group; character variety
Subject: Primary: 57M50; Secondary: 57M05, 20E36, 20F29

Abstract
Let (S, ∗) be a closed oriented surface with a marked point, let G be a fixed group, and let ρ:π1(S) ⟶ G be a representation such that the orbit of ρ under the action of the mapping class group Mod(S,∗) is finite. We prove that the image of ρ is finite. A similar result holds if π1(S) is replaced by the free group Fn on n≧ 2 generators, and where Mod(S,∗) is replaced by Aut(Fn). We show that if G is a linear algebraic group and if the representation variety of π1(S) is replaced by the character variety, then there are infinite image representations which are fixed by the whole mapping class group.

Acknowledgements

IB and MM acknowledge support of their respective J. C. Bose Fellowships. TK is partially supported by Simons Foundation Collaboration Grant number 429836, by an Alfred P. Sloan Foundation Research Fellowship, and by NSF Grant DMS-1711488.


Author information

Indranil Biswas:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
indranil@math.tifr.res.in

Thomas Koberda:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
thomas.koberda@gmail.com

Mahan Mj:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
mahan@math.tifr.res.in

Ramanujan Santharoubane:
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
ramanujan.santharoubane@gmail.com