Cycle decompositions and train tracks
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- by Charles A. Matthews and David J. Wright PDF
- Proc. Amer. Math. Soc. 132 (2004), 3411-3415 Request permission
Abstract:
We prove that the disjoint cycle decomposition of the permutation $(1 2 \cdots n_1)^{k_1} (1 2 \cdots n_2)^{k_2} \cdots (1 2 \cdots n_r)^{k_r}$ consists of cycles of at most $r$ distinct lengths. The proof relies on the geometry and topology of simple closed curves and train tracks on a closed surface of genus $r$.References
- Peter J. Cameron, Permutation groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, Cambridge, 1999. MR 1721031, DOI 10.1017/CBO9780511623677
- John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. MR 1409812, DOI 10.1007/978-1-4612-0731-3
- Andrew Haas and Perry Susskind, The connectivity of multicurves determined by integral weight train tracks, Trans. Amer. Math. Soc. 329 (1992), no. 2, 637–652. MR 1028309, DOI 10.1090/S0002-9947-1992-1028309-1
- R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770, DOI 10.1515/9781400882458
- William Thurston, The geometry and topology of $3$-Manifolds, Princeton University Press, Princeton, NJ, 1980.
Additional Information
- Charles A. Matthews
- Affiliation: Department of Mathematics, Southeastern Oklahoma State University, Durant, Oklahoma 74701
- Email: cmatthews@sosu.edu
- David J. Wright
- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74075
- Email: wrightd@math.okstate.edu
- Received by editor(s): February 18, 2002
- Received by editor(s) in revised form: November 10, 2002
- Published electronically: June 16, 2004
- Communicated by: Alan Dow
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 3411-3415
- MSC (2000): Primary 57N99, 20B30, 32G15, 30F99
- DOI: https://doi.org/10.1090/S0002-9939-04-07515-X
- MathSciNet review: 2073318