Some finite quotients of the mapping class group of a surface
HTML articles powered by AMS MathViewer
- by Patricia L. Sipe PDF
- Proc. Amer. Math. Soc. 97 (1986), 515-524 Request permission
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).References
- Michael F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62. MR 286136
- Clifford J. Earle, Families of Riemann surfaces and Jacobi varieties, Ann. of Math. (2) 107 (1978), no. 2, 255–286. MR 499328, DOI 10.2307/1971144 —, Roots of the canonical divisor class over Teichmüller space, preprint.
- Dennis Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2) 22 (1980), no. 2, 365–373. MR 588283, DOI 10.1112/jlms/s2-22.2.365
- W. B. R. Lickorish, A finite set of generators for the homeotopy group of a $2$-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769–778. MR 171269, DOI 10.1017/s030500410003824x
- David Mumford, Abelian quotients of the Teichmüller modular group, J. Analyse Math. 18 (1967), 227–244. MR 219543, DOI 10.1007/BF02798046
- C. L. Siegel, Topics in complex function theory. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Automorphic functions and abelian integrals; Translated from the German by A. Shenitzer and M. Tretkoff; With a preface by Wilhelm Magnus; Reprint of the 1971 edition; A Wiley-Interscience Publication. MR 1008931 P. L. Sipe, Roots of the canonical bundle of the universal Teichmüller curve, Thesis, Cornell Univ., 1979.
- Patricia L. Sipe, Roots of the canonical bundle of the universal Teichmüller curve and certain subgroups of the mapping class group, Math. Ann. 260 (1982), no. 1, 67–92. MR 664367, DOI 10.1007/BF01475756
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 97 (1986), 515-524
- MSC: Primary 32G15; Secondary 57N05
- DOI: https://doi.org/10.1090/S0002-9939-1986-0840639-5
- MathSciNet review: 840639