  Our main result is that the image of the quantum representation of a central
extension of the mapping class group of the genus $g\geq 3$ closed orientable
surface at a prime $p\geq 5$ is a Zariski dense discrete subgroup of some
higher rank algebraic semi-simple Lie group $\mathbb G_p$ defined over $\Q$. As
an application we find that, for any prime $p\geq 5$ a central extension of the
genus $g$ mapping class group surjects onto the finite groups $\mathbb
G_p(\Z/q\Z)$, 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.
