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The solvable length of groups of local diffeomorphisms

  • Javier Ribón EMAIL logo
Published/Copyright: December 14, 2016

Abstract

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we show that 2n+1 is the sharpest upper bound for the derived length of solvable subgroups of the group Diff(n,0) of local complex analytic diffeomorphisms for n=2,3,4,5.

Award Identifier / Grant number: 4389-14-0

Funding statement: Research partially funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, CAPES/DGU Postdoctoral fellowship 4389-14-0.

Acknowledgements

I thank the referee for the helpful suggestions improving the writing of the paper.

References

[1] A. Borel, Linear algebraic groups, 2nd ed., Grad. Texts in Math. 126, Springer, New York 1991. 10.1007/978-1-4612-0941-6Search in Google Scholar

[2] S. Cantat and D. Cerveau, Analytic actions of mapping class groups on surfaces, J. Topol. 1 (2008), no. 4, 910–922. 10.1112/jtopol/jtn028Search in Google Scholar

[3] D. Cerveau and J.-F. Mattei, Formes intégrables holomorphes singulières, Astérisque 97, Société Mathématique de France, Paris 1982. Search in Google Scholar

[4] D. Cerveau and R. Moussu, Groupes d’automorphismes de (𝐂,0) et équations différentielles ydy+=0, Bull. Soc. Math. France 116 (1988), no. 4, 459–488. 10.24033/bsmf.2108Search in Google Scholar

[5] L. J. Corwin and F. P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I: Basic theory and examples, Cambridge Stud. Adv. Math. 18, Cambridge University Press, Cambridge 1990. Search in Google Scholar

[6] J. D. Dixon, The solvable length of a solvable linear group, Math. Z. 107 (1968), 151–158. 10.1007/BF01111027Search in Google Scholar

[7] J. Écalle, Théorie itérative: Introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. (9) 54 (1975), 183–258. Search in Google Scholar

[8] J. Franks and M. Handel, Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface, J. Mod. Dyn. 7 (2013), no. 3, 369–394. 10.3934/jmd.2013.7.369Search in Google Scholar

[9] É. Ghys, Sur les groupes engendrés par des difféomorphismes proches de l’identité, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 2, 137–178. 10.1007/BF01237675Search in Google Scholar

[10] J. E. Humphreys, Linear algebraic groups, Grad. Texts in Math. 21, Springer, New York 1995. Search in Google Scholar

[11] Y. Ilyashenko and S. Yakovenko, Lectures on analytic differential equations, Grad. Stud. Math. 86, American Mathematical Society, Providence 2008. 10.1090/gsm/086Search in Google Scholar

[12] E. R. Kolchin, Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. (2) 49 (1948), 1–42. 10.2307/1969111Search in Google Scholar

[13] F. Loray, Cinq leçons sur la structure transverse d’une singularité de feuilletage holomorphe en dimension 2 complexe, Monographies Red TMR Europea Sing. Ec. Dif. Fol. 1 (1999), 1–92. Search in Google Scholar

[14] M. Martelo and J. Ribón, Derived length of solvable groups of local diffeomorphisms, Math. Ann. 358 (2014), no. 3, 701–728. 10.1007/s00208-013-0975-5Search in Google Scholar

[15] J. Martinet and J.-P. Ramis, Classification analytique des équations differentielles non linéaires résonnantes du premier ordre, Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 571–621. 10.24033/asens.1462Search in Google Scholar

[16] J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. Éc. Norm. Supér. (4) 13 (1980), no. 4, 469–523. 10.24033/asens.1393Search in Google Scholar

[17] J. J. Morales-Ruiz, J.-P. Ramis and C. Simo, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. Éc. Norm. Supér. (4) 40 (2007), no. 6, 845–884. 10.1016/j.ansens.2007.09.002Search in Google Scholar

[18] M. F. Newman, The soluble length of soluble linear groups, Math. Z. 126 (1972), 59–70. 10.1007/BF01580356Search in Google Scholar

[19] E. Paul, Feuilletages holomorphes singuliers à holonomie résoluble, J. reine angew. Math. 514 (1999), 9–70. 10.1515/crll.1999.074Search in Google Scholar

[20] J. C. Rebelo and H. Reis, Discrete orbits, recurrence and solvable subgroups of Diff(2,0), J. Geom. Anal. (2016), 10.1007/s12220-015-9671-x. 10.1007/s12220-015-9671-xSearch in Google Scholar

[21] J. Ribón, Recurrent orbits of subgroups of local complex analytic diffeomorphisms, Math. Z. (2016), 10.1007/s00209-016-1719-5. 10.1007/s00209-016-1719-5Search in Google Scholar

[22] J.-P. Serre, Lie algebras and Lie groups. 1964 lectures, given at Harvard University, 2nd ed., Lecture Notes in Math. 1500, Springer, Berlin 1992. 10.1007/978-3-540-70634-2_1Search in Google Scholar

[23] J. Shurman, Geometry of the quintic, John Wiley & Sons, New York 1997. Search in Google Scholar

[24] B. L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar Publishing, New York 1949. Search in Google Scholar

[25] B. A. F. Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergeb. Math. Grenzgeb. 76, Springer, New York 1973. 10.1007/978-3-642-87081-1Search in Google Scholar

[26] H. Zassenhaus, Beweis eines Satzes über diskrete Gruppen, Abh. Math. Sem. Univ. Hamburg 12 (1937), no. 1, 289–312. 10.1007/BF02948950Search in Google Scholar

[27] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funktsional. Anal. i Prilozhen. 16 (1982), no. 3, 30–41, 96. 10.1007/BF01081586Search in Google Scholar

[28] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II, Funktsional. Anal. i Prilozhen. 17 (1983), no. 1, 8–23. 10.1007/BF01083174Search in Google Scholar

Received: 2015-09-07
Revised: 2016-10-05
Published Online: 2016-12-14
Published in Print: 2019-07-01

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