论文标题
符号组的有限商与映射课程组
Finite quotients of symplectic groups vs mapping class groups
论文作者
论文摘要
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.我们还可以以这种方式检索deligne的非残基有限性$ \ wideTilde {sp（2g，\ mathbb z）} $。然后，我们证明，第二个同源性在算术类型的Dedekind域中，第二个同源物的图像是均匀界限大小的扭转组。相比之下，量子表示为每一个prime $ p $生产，这是$ g \ geq 3 $映射类的有限商，其第二个同源图像具有$ p $ torsion。我们进一步得出，映射类组的所有中心扩展都是残留有限的，并推断出映射类组具有SERRE的属性$ A_2 $用于琐碎模块，这与符号组相反。最终，我们计算共同变量的模块$ h_2（\ mathfrak {sp} _ {2G}（2））_ {sp（2G，\ Mathbb z/2^k \ mathbb z）= \ Mathbb z/2 \ zybb z/2 \ mathbb z $。
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 $H_2(\mathfrak{sp}_{2g}(2))_{Sp(2g,\mathbb Z/2^k\mathbb Z)}=\mathbb Z/2\mathbb Z$.
