Geometriae Dedicata

, Volume 183, Issue 1, pp 195–213 | Cite as

Affine automorphisms of rooted trees

Original Paper

Abstract

We introduce a class of automorphisms of rooted d-regular trees arising from affine actions on their boundaries viewed as infinite dimensional modules \({\mathbb {Z}}_d^{\infty }\). This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from this class and show that it is isomorphic to the index 2 extension of the rank 2 lamplighter group \({\mathbb {Z}}_2^2\wr {\mathbb {Z}}\).

Keywords

Automorphisms of rooted trees Self-similar groups Affine actions Lamplighter group Automata groups 

Mathematics Subject Classification

20E08 20F65 

References

  1. 1.
    Bondarenko, I., D’Angeli, D., Rodaro, E.: The lamplighter group \(\mathbb{Z}_3\wr \mathbb{Z}\) generated by a bireversible automaton. Preprint: arXiv:1502.07981 (2015)
  2. 2.
    Brunner, A.M., Sidki, S.: On the automorphism group of the one-rooted binary tree. J. Algebra 195(2), 465–486 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bartholdi, L.I., Šuniḱ, Z.: Some solvable automaton groups. In: Grigorchuk, R., Mihalik, M., Sapir, M., Šuniḱ, Z. (eds.) Topological and Asymptotic Aspects of Group Theory, vol. 394 of Contemp. Math., pp. 11–29. American Mathematical Society, Providence (2006)Google Scholar
  4. 4.
    Berlatto, A., Sidki, S.: Virtual endomorphisms of nilpotent groups. Groups Geom. Dyn. 1(1), 21–46 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brunner, A.M., Sidki, S.N.: Abelian state-closed subgroups of automorphisms of \(m\)-ary trees. Groups Geom. Dyn. 4(3), 455–472 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Caponi, L.: On Classification of Groups Generated by Automata with 4 States over a 2-Letter Alphabet. Master’s thesis, University of South Florida, Department of Mathematics and Statistics, Tampa, FL, 33620, USA (2014)Google Scholar
  7. 7.
    Dantas, A., Sidki, S.: On state-closed representations of restricted wreath product of groups of type \({G}_{p,d}={C}_{p} wr {C}^{d}\). Preprint: arXiv:1505.05165 (2015)
  8. 8.
    Grigorchuk, R., Kravchenko, R.: On the lattice of subgroups of the lamplighter group. Int. J. Algebra Comput. 24(6), 837–877 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Grigorchuk, R.I., Nekrashevich, V.V., Sushchanskiĭ, V.I.: Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231(Din. Sist., Avtom. i Beskon. Gruppy):134–214 (2000)Google Scholar
  10. 10.
    Grigorchuk, R., Savchuk, D.: Self-similar groups acting essentially freely on the boundary of the binary rooted tree. In: Group theory, combinatorics, and computing, vol 611 of Contemp. Math., pp. 9–48. American Mathematical Society, Providence (2014)Google Scholar
  11. 11.
    Grigorchuk, R.I., Żuk, A.: The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedic. 87(1–3), 209–244 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Klimann, I., Picantin, M., Savchuk, D.: Orbit automata as a new tool to attack the order problem in automaton groups. J. Algebra 445, 433–457 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Muntyan, Y., Savchuk, D.: AutomGrp—GAP Package for Computations in Self-Similar Groups and Semigroups, Version 1.2.4. http://www.gap-system.org/Packages/automgrp.html (2014)
  14. 14.
    Nekrashevych, V.: Self-Similar Groups, vol. 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2005)Google Scholar
  15. 15.
    Nekrashevych, V., Sidki, S.: Automorphisms of the Binary Tree: State-Closed Subgroups and Dynamics of \(1/2\)-Endomorphisms, vol. 311 of London Math. Soc. Lect. Note Ser., pp. 375–404. Cambridge University Press, Cambridge (2004)Google Scholar
  16. 16.
    Silva, P.V., Steinberg, B.: On a class of automata groups generalizing lamplighter groups. Int. J. Algebra Comput. 15(5–6), 1213–1234 (2005)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Woryna, A.: The concept of self-similar automata over a changing alphabet and lamplighter groups generated by such automata. Theor. Comput. Sci. 412, 96–110 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA
  2. 2.Departamneto de MatematicaUniversidade de BrasíliaBrasíliaBrazil

Personalised recommendations