Finite quotients of symplectic groups vs mapping class groups

We prove that the essential second homology of finite quotients of symplectic groups over a fixed Dedekind domain of arithmetic type are torsion groups of uniformly bounded size. This is shown to be equivalent to a result of Deligne proving that suitable central extensions of higher rank Chevalley groups over Dedekind domains of arithmetic type are not residually finite. Using this equivalence one shows that Deligne's non-residual finiteness result is sharp by showing that homomorphisms of the universal central extension of $Sp(2g,\Z)$ to finite groups factor through some nontrivial extension of $Sp(2g,\Z)$ by $\Z/2\Z$, when $g\geq 3$. We provide several proofs of this statement, using elementary calculations, K-theory arguments based on the work of Barge and Lannes or Weil representations of symplectic groups arising in abelian Chern-Simons theory. Furthermore, one proves that, for every prime $p\geq 5$, the mapping class group of genus $g\geq 3$ has finite quotients whose essential second homology has $p$-torsion. This is a simple consequence of the existence of quantum representations associated to SU(2) topological quantum field theory in dimension three.