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Finite generation of iterated wreath products

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Abstract

Let (G n , X n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product \({\ldots\wr G_2\wr G_1}\) is topologically finitely generated if and only if the profinite abelian group \({\prod_{n\geq 1} G_n/G'_n}\) is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power \({G\wr \ldots\wr G\wr G}\) (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.

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References

  1. Bartholdi L.: A Wilson group of non-uniformly exponential growth. C. R. Math. Acad. Sci. Paris 336, 549–554 (2003)

    MATH  MathSciNet  Google Scholar 

  2. L. Bartholdi, R. Grigorchuk, and Z. Šuniḱ, Branch groups, in: Handbook of algebra, vol. 3, pp. 989–1112. North-Holland, Amsterdam, 2003.

  3. Bhattacharjee M.: The probability of generating certain profinite groups by two elements. Isr. J. Math. 86, 311–329 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. David C.: Generating numbers for wreath products. Rend. Semin. Mat. Univ. Padova 92, 71–77 (1994)

    MATH  Google Scholar 

  5. Grigorchuk R.: Branch groups. Math. Notes 67, 718–723 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Grigorchuk, Solved and unsolved problems around one group, in: Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math. 248, pp. 117–218. Birkhäuser, Basel, 2005.

  7. Lucchini A.: Generating wreath products and their augmentation ideals. Rend. Semin. Mat. Univ. Padova 98, 67–87 (1997)

    MATH  MathSciNet  Google Scholar 

  8. Neumann P.: Some questions of Edjvet and Pride about infinite groups. Ill. J. Math. 30, 301–316 (1986)

    MATH  Google Scholar 

  9. Quick M.: Probabilistic generation of wreath products of non-Abelian finite simple groups. Commun. Algebra 32, 4753–4768 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Quick M.: Probabilistic generation of wreath products of non-Abelian finite simple groups. II. Int. J. Algebra Comput. 16, 493–503 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Segal D.: The finite images of finitely generated groups. Proc. Lond. Math. Soc. (3) 82, 597–613 (2001)

    Article  MATH  Google Scholar 

  12. Wiegold J.: Growth sequences of finite groups II. J. Austral. Math. Soc. 20, 225–229 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wiegold J., Wilson J.: Growth sequences of finitely generated groups. Arch. Math. 30, 337–343 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  14. Woryna A.: On generation of wreath products of cyclic groups by two state time varying Mealy automata. Int. J. Algebra Comput. 16, 397–415 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Mechanics and Mathematics Department, National Taras Shevchenko University of Kyiv, vul.Volodymyrska 64, Kiev, 01033, Ukraine

    Ievgen V. Bondarenko

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  1. Ievgen V. Bondarenko
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Bondarenko, I.V. Finite generation of iterated wreath products. Arch. Math. 95, 301–308 (2010). https://doi.org/10.1007/s00013-010-0169-2

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