 

Indranil Biswas, Thomas Koberda, Mahan Mj, and Ramanujan Santharoubane
Representations of surface groups with finite mapping class group orbits view print


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:π_{1}(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 π_{1}(S) is replaced by the free group F_{n} on n≧ 2
generators, and where Mod(S,∗) is replaced by Aut(F_{n}). We show that if G is a linear algebraic group and if the representation variety of π_{1}(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 DMS1711488.


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 229044137, 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 229044137, USA
ramanujan.santharoubane@gmail.com

