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 Mosc. Math. J., 2002, Volume 2, Number 1, Pages 127–160 (Mi mmj49)

Orbits of braid groups on cacti

G. A. Jonesa, A. K. Zvonkinb

a University of Southampton
b Universite Bordeaux 1, Laboratoire Bordelais de Recherche en Informatique

Abstract: One of the consequences of the classification of finite simple groups is the fact that non-rigid polynomials (those with more than two finite critical values), considered as branched coverings of the sphere, have exactly three exceptional monodromy groups (one in degree 7, one in degree 13 and one in degree 15). By exceptional here we mean primitive and not equal to $S_n$ or $A_n$, where $n$ is the degree. Motivated by the problem of the topological classification of polynomials, a problem that goes back to 19th century researchers, we discuss several techniques for investigating orbits of braid groups on “cacti” (ordered sets of monodromy permutations). Applying these techniques, we provide a complete topological classification for the three exceptional cases mentioned above.

Key words and phrases: Topological classification of polynomials, monodromy groups, Braid group actions.

DOI: https://doi.org/10.17323/1609-4514-2002-2-1-127-160

Full text: http://www.ams.org/.../abst2-1-2002.html
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Bibliographic databases:

MSC: Primary 30C10; Secondary 57M12, 05B25, 57M60, 20B15
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Citation: G. A. Jones, A. K. Zvonkin, “Orbits of braid groups on cacti”, Mosc. Math. J., 2:1 (2002), 127–160

Citation in format AMSBIB
\Bibitem{JonZvo02} \by G.~A.~Jones, A.~K.~Zvonkin \paper Orbits of braid groups on cacti \jour Mosc. Math.~J. \yr 2002 \vol 2 \issue 1 \pages 127--160 \mathnet{http://mi.mathnet.ru/mmj49} \crossref{https://doi.org/10.17323/1609-4514-2002-2-1-127-160} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1900588} \zmath{https://zbmath.org/?q=an:1008.20030} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208587700008} \elib{http://elibrary.ru/item.asp?id=8379099} 

• http://mi.mathnet.ru/eng/mmj49
• http://mi.mathnet.ru/eng/mmj/v2/i1/p127

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Jones G.A., “Cyclic regular subgroups of primitive permutation groups”, Journal of Group Theory, 5:4 (2002), 403–407
2. Kwak J.H., Lee J., Mednykh A., “Coverings, enumeration and Hurwitz problems”, Applications of Group Theory to Combinatorics, 2008, 71–107
3. Mueller P., “Permutation Groups with a Cyclic Two-Orbits Subgroup and Monodromy Groups of Laurent Polynomials”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 12:2 (2013), 369–438
4. James A., Magaard K., Shpectorov S., “the Lift Invariant Distinguishes Components of Hurwitz Spaces For a(5)”, Proc. Amer. Math. Soc., 143:4 (2015), PII S0002-9939(2014)12185-X, 1377–1390
5. J. Math. Sci. (N. Y.), 226:5 (2017), 548–560