Low‐dimensional linear representations of mapping class groups

Abstract

AbstractLet be a compact orientable surface of genus 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 of to is trivial if and is finite cyclic if . The first result is our own proof of this fact. Our second main result shows that for up to conjugation there are only two homomorphisms from to : 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 . We provide many applications of our results, including the finiteness of homomorphisms from mapping class groups of nonorientable surfaces to , the triviality of homomorphisms from the mapping class groups to or to , and homomorphisms between mapping class groups. We also show that if the surface has marked point but no boundary components, then is generated by involutions if and only if and .