Finite quotients of symplectic groups vs mapping class groups

Abstract

We give alternative computations of the Schur multiplier of \(Sp(2g,\mathbb{Z}/D\mathbb{Z})\), when \(D\) is divisible by 4 and \(g\geq 4\): a first one using \(K\)-theory arguments based on the work of Barge and Lannes and a second one based on the Weil representations of symplectic groups arising in abelian Chern-Simons theory. We can also retrieve this way Deligne's non-residual finiteness of the universal central extension \(\widetilde{Sp(2g,\mathbb{Z})}\). We prove then that the image of the second homology into finite quotients of symplectic groups over a Dedekind domain of arithmetic type are torsion groups of uniformly bounded size. In contrast, quantum representations produce for every prime \(p\), finite quotients of the mapping class group of genus \(g\geq 3\) whose second homology image has \(p\)-torsion. We further derive that all central extensions of the mapping class group are residually finite and deduce that mapping class groups have Serre's property \(A_2\) for trivial modules, contrary to symplectic groups. Eventually we compute the module of coinvariants \(H_2(\mathfrak{sp}_{2g}(2))_{Sp(2g,\mathbb{Z}/2^k\mathbb{Z})}=\mathbb{Z}/2\mathbb{Z}\).