Transformation Groups

, Volume 15, Issue 3, pp 577–610 | Cite as

Normal subgroup generated by a plane polynomial automorphism



We study the normal subgroup 〈f N generated by an element f ≠ id in the group G of complex plane polynomial automorphisms having Jacobian determinant 1. On the one hand, if f has length at most 8 relative to the classical amalgamated product structure of G, we prove that 〈f N = G. On the other hand, if f is a sufficiently generic element of even length at least 14, we prove that 〈f N G.


Normal Subgroup Commutator Subgroup Primary Vertex Serre Tree Initial Vertex 
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  1. [Ani83]
    D. J. Anick, Limits of tame automorphisms of k[x 1,…,x N], J. Algebra 82 (1983), no. 2, 459–468.CrossRefMathSciNetMATHGoogle Scholar
  2. [Bla10]
    J. Blanc, Groupes de Cremona, connexité et simplicité, Ann. Sci. École Norm. Sup. (2010), to appear.Google Scholar
  3. [CD08]
    D. Cerveau, J. Deserti, Transformations birationnelles de petit degré, arXiv: 0811.2325 (2008).Google Scholar
  4. [Dan74]
    В. И. Данилов, Неnросmоmа груnnы унимодулярных авmоморфизмов аффинной nлоскосmи, Мат. Заметки 15 (1974), 289–293. Engl. transl.: V. I. Danilov, Nonsimplicity of the group of unimodular automorphisms of the affine plane, Math. Notes 15 (1974), 165–167.Google Scholar
  5. [FM89]
    S. Friedland, J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 67–99.CrossRefMathSciNetMATHGoogle Scholar
  6. [Fur07]
    J.-P. Furter, Jet groups, J. Algebra 315 (2007), no. 2, 720–737.CrossRefMathSciNetMATHGoogle Scholar
  7. [Giz94]
    M. H. Gizatullin, The decomposition, inertia and ramification groups in birational geometry, in: Algebraic Geometry and its Applications (Yaroslavl’, 1992), Aspects of Mathematics, E25, Vieweg, Braunschweig, 1994, pp. 39–45.Google Scholar
  8. [Isk85]
    В. А. Исковских, Доказательство теоремы о соотношениях в двумерной груnnе Кремоны, УМН 40 (1985), no. 5(245), 255–256. Engl. transl.: V. A. Iskovskikh, Proof of a theorem on relations in the two-dimensional Cremona group, Russ. Math. Surv. 40 (1985), no. 5, 231–232.Google Scholar
  9. [Jun42]
    H. W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161–174.MathSciNetGoogle Scholar
  10. [Kam96]
    T. Kambayashi, Pro-affine algebras, ind-affine groups and the Jacobian problem, J. Algebra 185 (1996), no. 2, 481–501.CrossRefMathSciNetMATHGoogle Scholar
  11. [Kam03]
    T. Kambayashi, Some basic results on pro-affine algebras and ind-affine schemes, Osaka J. Math. 40 (2003), no. 3, 621–638.MathSciNetMATHGoogle Scholar
  12. [Lam01]
    S. Lamy, L'alternative de Tits pour \({\rm Aut} [\mathbb{C}^{2}]\), J. Algebra 239 (2001), no. 2, 413–437.CrossRefMathSciNetMATHGoogle Scholar
  13. [Lam02]
    S. Lamy, Une preuve géométrique du théorème de Jung, Enseign. Math. (2) 48 (2002), nos. 3–4, 291–315.MathSciNetMATHGoogle Scholar
  14. [LS01]
    R. C. Lyndon, P. E. Schupp, Combinatorial Group Theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. Russian transl.: Р. Линдон, П. Шупп, Комбинаторная теория груnn, Мир, М. 1980.Google Scholar
  15. [Lyn66]
    R. C. Lyndon, On Dehn's algorithm, Math. Ann. 166 (1966), 208–228.CrossRefMathSciNetMATHGoogle Scholar
  16. [Mau01]
    S. Maubach, Polynomial automorphisms over finite fields, Serdica Math. J. 27 (2001), no. 4, 343–350.MathSciNetMATHGoogle Scholar
  17. [Nag72]
    M. Nagata, On automorphism group of k[x, y], Department of Mathematics, Kyoto University, Lectures in Mathematics, No. 5, Kinokuniya Book-Store, Tokyo, 1972.Google Scholar
  18. [Rot95]
    J. J. Rotman, An Introduction to the Theory of Groups, 4th ed., Graduate Texts in Mathematics, Vol. 148, Springer-Verlag, New York, 1995.Google Scholar
  19. [Sch71]
    P. E. Schupp, Small cancellation theory over free products with amalgamation, Math. Ann. 193 (1971), 255–264.CrossRefMathSciNetGoogle Scholar
  20. [Ser77]
    J.-P. Serre, Arbres, Amalgames, SL2, Société Mathématique de France, Paris, 1977. Avec un sommaire anglais. Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46.Google Scholar
  21. [Sha66]
    I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. Appl. (5) 25 (1966), nos. 1–2, 208–212.MathSciNetGoogle Scholar
  22. [Sha81]
    И. Р. Шафаревич, О некоторых бесконетерных груnnах, II, Изв. АН ССР, сер. мат. 45 (1981), no. 1, 214–226, 240. Engl. transl.: I. R. Shafarevich, On some infinite-dimensional groups, II, Math. USSR, Izv. 18 (1982), 185–194.Google Scholar
  23. [vdK53]
    W. van der Kulk, On polynomial rings in two variables, Nieuw Arch. Wisk. (3) 1 (1953), 33–41.MATHGoogle Scholar
  24. [VK33]
    E. R. Van Kampen, On the connection between the fundamental groups of some related spaces, Amer. J. Math. 55 (1933), 261–267.Google Scholar
  25. [Wei66]
    C. M. Weinbaum, Visualizing the word problem, with an application to sixth groups, Pacific J. Math. 16 (1966), 557–578.MathSciNetMATHGoogle Scholar
  26. [Wri79]
    D. Wright, Abelian subgroups of Autk(k[X, Y]) and applications to actions on the affine plane, Illinois J. Math. 23 (1979), no. 4, 579–634.MathSciNetMATHGoogle Scholar
  27. [Wri92]
    D. Wright, Two-dimensional Cremona groups acting on simplicial complexes, Trans. Amer. Math. Soc. 331 (1992), no. 1, 281–300.CrossRefMathSciNetMATHGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Laboratoire MIAUniversité de La RochelleLa Rochelle cedexFrance
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUnited Kingdom

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