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 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 π₁(S) is replaced by the character variety, then there are infinite image representations which are fixed by the whole mapping class group.

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.