Low-dimensional linear representations of mapping class groups

Let S$S$ be a compact orientable surface of genus g$g$ with marked points in the interior. Franks-Handel (Proc. Amer. Math. Soc. 141 (2013) 2951-2962) proved that if n<2g$n<2g$ then the image of a homomorphism from the mapping class group Mod(S)${\rm Mod}(S)$ of S$S$ to GL(n,C)${\rm GL}(n,{\mathbb {C}})$ is trivial if g & GT;3$g\geqslant 3$ and is finite cyclic if g=2$g=2$. The first result is our own proof of this fact. Our second main result shows that for g & GT;3$g\geqslant 3$ up to conjugation there are only two homomorphisms from Mod(S)${\rm Mod}(S)$ to GL(2g,C)${\rm GL}(2g,{\mathbb {C}})$: the trivial homomorphism and the standard symplectic representation. Our last main result shows that the mapping class group has no faithful linear representation in dimensions less than or equal to 3g-3$3g-3$. We provide many applications of our results, including the finiteness of homomorphisms from mapping class groups of nonorientable surfaces to GL(n,C)${\rm GL}(n,{\mathbb {C}})$, the triviality of homomorphisms from the mapping class groups to Aut(Fn)${\rm Aut}(F_n)$ or to Out(Fn)${\rm Out}(F_n)$, and homomorphisms between mapping class groups. We also show that if the surface S$S$ has r$r$ marked point but no boundary components, then Mod(S)${\rm Mod}(S)$ is generated by involutions if and only if g & GT;3$g\geqslant 3$ and r & LE;2g-2$r\leqslant 2g-2$.