Some finite quotients of the mapping class group of a surface

Abstract:

Let $S$ be a smooth, oriented, compact surface of genus $p \geqslant 2$, and ${\operatorname {Mod} _p}$ its Teichmüller modular group (or mapping class group). Let ${T_1}(S)$ denote the unit tangent bundle, and let $n$ be an integer dividing $2p - 2$. ${\operatorname {Mod} _p}$ acts on the finite set ${\Phi _n}$, the elements of which are certain homomorphisms from ${H_1}({T_1}(S),{{\mathbf {Z}}_n})$ to ${{\mathbf {Z}}_n}$. In previous work of the author, these homomorphisms arose as the topological description of the $n$th roots of the canonical bundle of the universal Teichmüller curve; however, a topological approach is taken here. The subgroups of ${G_{p,n}}$ which leave all elements of ${\Phi _n}$ fixed are subgroups of finite index in ${\operatorname {Mod} _p}$. Let ${Q_n} = {\operatorname {Mod} _p}/{G_{p,n}}$. The elements of ${Q_n}$ are characterized algebraically. ${Q_n}$ is an extension of ${(2{{\mathbf {Z}}_n})^{2p}}$ by the symplectic group ${\text {Sp(p,}}{{\mathbf {Z}}_n})$ (and in the case of $n \operatorname {odd}, {Q_n}$ is a semidirect product).