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Cycle decompositions and train tracks

Author(s): Charles A. Matthews; David J. Wright
Journal: Proc. Amer. Math. Soc. 132 (2004), 3411-3415.
MSC (2000): Primary 57N99, 20B30, 32G15, 30F99
Posted: June 16, 2004
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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 distinct lengths. The proof relies on the geometry and topology of simple closed curves and train tracks on a closed surface of genus .

References:

1.: Peter J. Cameron, Permutation groups, London Mathematical Society Student Texts, no. 45, Cambridge University Press, Cambridge, 1999. MR 2001c:20008
2.: John D. Dixon and Brian Mortimer, Permutation groups, Graduate Texts in Mathematics, no. 163, Springer-Verlag, New York, 1996. MR 98m:20003
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 92e:57024
4.: R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Math. Studies, no. 125, Princeton University Press, Princeton, NJ, 1992. MR 94b:57018
5.: William Thurston, The geometry and topology of -Manifolds, Princeton University Press, Princeton, NJ, 1980.


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