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A minimal Cantor set in the space of 3-generated groups

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Abstract

We construct and study a family of 3-generated groups \(\mathcal{D}_w\) parametrized by infinite binary sequences w. We show that two groups of the family are isomorphic if and only if the sequences are cofinal and that two groups cannot be distinguished by finite sets of relations. We show a connection of the family with 2-dimensional holomorphic dynamics.

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References

  1. Bartholdi, L., Grigorchuk, R.I., Šuniḱ, Z.: Branch groups. In: Handbook of Algebra, Vol. 3 pp. 989–1112 North-Holland, Amsterdam (2003)

  2. Bartholdi, L., Nekrashevych, V.V.: Thurston equivalence of topological polynomials. (to appear in “Acta Math.”, available at http://www.math.tamu.edu/ nekrash/Preprints/rabbit.pdf) (2005)

  3. Champetier C. (2000). L’espace de groupes de type fini. Topology 39: 657–680

    Article  MATH  MathSciNet  Google Scholar 

  4. Erschler A. (2004). Boundary behaviour for groups of subexponential growth. Ann. Math. 160: 1183–1210

    Article  MathSciNet  Google Scholar 

  5. Fornæss, J.E., Sibony, N.: ‘Critically finite rational maps on \(\mathbb{P}^2\)’. In: The Madison Symposium on Complex Analysis (Madison, WI, 1991), Vol. 137. pp. 245–260. of Contemp. Math. Amer. Math. Soc. Providence, RI (1992)

  6. Gawron P.W., Nekrashevych V.V. and Sushchansky V.I. (2001). Conjugation in tree automorphism groups. Int. J. Algebra Comput. 11(5): 529–547

    Article  MATH  MathSciNet  Google Scholar 

  7. Grigorchuk, R.: ‘Solved and unsolved problems aroud one group’. In: Bartholdi, T.S.-N.L., Ceccherini-Silberstein, T., Żuk, A.(eds.) Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Vol. 248, pp. 117–218. Of Progress in Mathematics. Birkhäuser, Basel (2005)

  8. Grigorchuk R.I. (1985). Degrees of growth of finitely generated groups and the theory of invariant means. Math. USSR Izv. 25(2): 259–300

    Article  MATH  MathSciNet  Google Scholar 

  9. Lavreniuk Y.V. and Nekrashevych V.V. (2002). Rigidity of branch groups acting on rooted trees. Geom. Dedicata 89(1): 155–175

    Article  MathSciNet  Google Scholar 

  10. Nekrashevych, V.: Self-similar groups, Vol. 117 of Mathematical Surveys and Monographs. Amer. Math. Soc., Providence, RI (2005)

  11. Nekrashevych, V.: An uncountable family of three-generated groups acting on the binary tree. (preprint, available at http://www.math.tamu.edu/ nekrash/Preprints/mixed.pdf) (2005)

  12. Nekrashevych, V., Sidki, S.: Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms’. In: Müller, T.W. (ed.) Groups: Topological, Combinatorial and Arithmetic Aspects, Vol. 311 of LMS Lecture Notes Series. Cambridge University Press, Cambridge. pp. 375–404 (2004)

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  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Volodymyr Nekrashevych

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  1. Volodymyr Nekrashevych
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Nekrashevych, V. A minimal Cantor set in the space of 3-generated groups. Geom Dedicata 124, 153–190 (2007). https://doi.org/10.1007/s10711-006-9118-4

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  • DOI: https://doi.org/10.1007/s10711-006-9118-4

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