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Geometriae Dedicata

, Volume 163, Issue 1, pp 339–348 | Cite as

Finite-state self-similar actions of nilpotent groups

  • Ievgen V. Bondarenko
  • Rostyslav V. Kravchenko
Original Paper

Abstract

Let G be a finitely generated torsion-free nilpotent group and \({\phi:H\rightarrow G}\) be a surjective homomorphism from a subgroup H < G of finite index with trivial \({\phi}\) -core. For every choice of coset representatives of H in G there is a faithful self-similar action of the group G associated with \({(G, \phi)}\). We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for \({(G, \phi)}\). These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism \(\widehat{\phi}\) of the Lie algebra of the Mal’cev completion of G.

Keywords

Self-similar action Nilpotent group Finite-state action Automaton group 

Mathematics Subject Classification (2000)

20F65 20F18 

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References

  1. 1.
    Bartholdi L., Sunik Z.: Some solvable automaton groups. Contemp. Math. 394, 11–29 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berlatto A., Sidki S.: Virtual endomorphisms of nilpotent groups. Groups Geom. Dyn. 1(1), 21–46 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brunner A.M., Sidki S.: The generation of \({GL(n,\mathbb{Z})}\) by finite state automata. Int. J. Algebra Comput. 8(1), 127–139 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Corwin, L.J., Greenleaf, F.P.: Representations of nilpotent Lie groups and their applications. Part 1: Basic theory and examples. Cambridge Studies in Advanced Mathematics, vol. 18. Cambridge University Press, Cambridge (1990)Google Scholar
  5. 5.
    Grigorchuk R.I., Nekrashevych V.V., Sushchansky V.I.: Automata, dynamical systems and groups. Proc. Steklov Inst. Math. 231, 128–203 (2000)Google Scholar
  6. 6.
    Grigorchuk R.I., A.: The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata 87(1–3), 209–244 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kapovich, M.: Arithmetic aspects of self-similar groups. Groups Geom. Dyn. (To appear)Google Scholar
  8. 8.
    Malcev A.I.: On a class of homogeneous spaces. Izv. Akad. Nauk SSSR Ser. Mat. 13, 9–32 (1949)MathSciNetGoogle Scholar
  9. 9.
    Nekrashevych, V.: Self-similar groups. Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005)Google Scholar
  10. 10.
    Nekrashevych, V., Sidki, S.: Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms. In: Muller, T.W. (ed.) Groups: Topological, Combinatorial and Arithmetic Aspects, vol. 311 of LMS Lecture Notes Series, pp. 375–404 (2004)Google Scholar
  11. 11.
    Savchuk D., Vorobets Y.: Automata generating free products of groups of order 2. J. Algebra 336(1), 53–66 (2011)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Warfield R.B.: Nilpotent groups. Lecture Notes in Mathematics, vol. 513. Springer, Berlin-Heidelberg-New York (1976)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Ievgen V. Bondarenko
    • 1
  • Rostyslav V. Kravchenko
    • 2
  1. 1.Mechanics and Mathematics DepartmentNational Taras Shevchenko University of KyivKyivUkraine
  2. 2.Department of MathematicsUniversity of ChicogoChicagoUSA

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