Low-dimensional linear representations of mapping class groups

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