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Some finite quotients of the mapping class group of a surface


Author: Patricia L. Sipe
Journal: Proc. Amer. Math. Soc. 97 (1986), 515-524
MSC: Primary 32G15; Secondary 57N05
MathSciNet review: 840639
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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).


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DOI: https://doi.org/10.1090/S0002-9939-1986-0840639-5
Article copyright: © Copyright 1986 American Mathematical Society