Zariski Density and Finite Quotients of Mapping Class Groups

Our main result is that the image of the quantum representation of a central extension of the mapping class group of the genus g≥3 closed orientable surface at a prime p≥5 is a Zariski dense discrete subgroup of some higher rank algebraic semi-simple Lie group $\mathbb G_p$ defined over $\mathbb Q$. As an application, we find that, for any prime p≥5, a central extension of the genus g mapping class group surjects onto the finite groups $\mathbb G_p(\mathbb Z/q\mathbb Z)$, for all but finitely many primes q. This method provides infinitely many finite quotients of a given mapping class group outside the realm of symplectic groups.