Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups

We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the SU(n)-Verlinde bundles over Teichmuller space is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus of the underlying surface is at least 3. For the genus 2 case, this intersection is exactly the order two subgroup, generated by the hyper-elliptic involution, in the case of even degree and $n=2$. Otherwise the intersection is also trivial in the genus 2 case.