Let $M=M(g)$ be the mapping class group of a surface of genus $g > 1$ (resp., $M = \mathrm{Aut}(F_g)$, the automorphism group of the free group on $g$ generators). As it is well known, $M$ is mapped onto the symplectic group $Sp(2g , \mathbb{Z})$ (resp., the general linear group $GL(g , \mathbb{Z})$). We will show that this is only a first case in a series: in fact, for every pair $(S , r)$ when S is a finite group with less than g generators and $r$ is a $\mathbb{Q}$-irreducible representation of $S$, we associate an arithmetic group which is then shown to be a virtual quotient of $M$. The case when $S$ is the trivial group gives the above $Sp(2g , \mathbb{Z})$ (resp., $GL(g , \mathbb{Z})$) but many new quotients are obtained. For example it is used to show that $M(2)$ (resp., $\mathrm{Aut}(F_3)$) is virtually mapped onto a non-abelian free group. Another application is an answer to a question of Kowalski: generic elements in the Torelli groups are hyperbolic and fully irreducible.

Joint work with Fritz Gruenwald, Michael Larsen and Justin Malestein.