Skip to main content
SpringerLink
Log in
SpringerLink
Log in

A Connected 3-State Reversible Mealy Automaton Cannot Generate an Infinite Burnside Group

  • Conference paper
  • First Online:
Developments in Language Theory (DLT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9168))

Included in the following conference series:

Abstract

The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, no such examples have been constructed in some classes, as groups generated by non reversible automata. It was recently shown that 2-state reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state reversible Mealy automata, using new original techniques. The results rely on a fine analysis of associated orbit trees and a new characterization of the existence of elements of infinite order.

This work was partially supported by the French Agence Nationale de la Recherche, through the Project MealyM ANR-JS02-012-01. The third author was partially supported by the New Researcher Grant from the USF Internal Awards Program.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akhavi, A., Klimann, I., Lombardy, S., Mairesse, J., Picantin, M.: On the finiteness problem for automaton (semi)groups. Int. J. Algebr. Comput. 22(6) (2012)

    Google Scholar 

  2. Alešin, S.V.: Finite automata and the Burnside problem for periodic groups. Mat. Zametki 11, 319–328 (1972)

    MathSciNet  Google Scholar 

  3. Bartholdi, L.: FR functionally recursive groups, self-similar groups, GAP package for computation in self-similar groups and semigroups, V. 2.2.1 (2015)

    Google Scholar 

  4. Bartholdi, L., Šuniḱ, Z.: On the word and period growth of some groups of tree automorphisms. Comm. Algebra 29–11, 4923–4964 (2001)

    Article  Google Scholar 

  5. Belk, J., Bleak, C.: Some undecidability results for asynchronous transducers and the Brin-Thompson group \(2{V}\). ArXiv:1405.0982

  6. Burnside, W.: On an unsettled question in the theory of discontinuous groups. Quart. J. Math. 33, 230–238 (1902)

    Google Scholar 

  7. D’Angeli, D., Rodaro, E.: Freeness of automata groups vs boundary dynamics. ArXiv:1410.6097v2

  8. D’Angeli, D., Rodaro, E.: A geometric approach to (semi)-groups defined by automata via dual transducers. Geometriae Dedicata 174(1), 375–400 (2015)

    Article  MathSciNet  Google Scholar 

  9. The GAP Group: GAP Groups, Algorithms, and Programming (2015)

    Google Scholar 

  10. Gawron, P.W., Nekrashevych, V.V., Sushchansky, V.I.: Conjugation in tree automorphism groups. Internat. J. Algebr. Comput. 11–5, 529–547 (2001)

    Article  MathSciNet  Google Scholar 

  11. Gillibert, P.: The finiteness problem for automaton semigroups is undecidable. Internat. J. Algebr. Comput. 24–1, 1–9 (2014)

    Article  MathSciNet  Google Scholar 

  12. Gluškov, V.M.: Abstract theory of automata. Uspehi Mat. Nauk 16(5), 3–62 (1961)

    MathSciNet  Google Scholar 

  13. Godin, T., Klimann, I., Picantin, M.: On Torsion-Free Semigroups Generated by Invertible Reversible Mealy Automata. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 328–339. Springer, Heidelberg (2015)

    Google Scholar 

  14. Golod, E.S., Shafarevich, I.: On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28, 261–272 (1964)

    MathSciNet  Google Scholar 

  15. Grigorchuk, R.: On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen 14(1), 53–54 (1980)

    Article  MathSciNet  Google Scholar 

  16. Grigorchuk, R.: Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48(5), 939–985 (1984)

    MathSciNet  Google Scholar 

  17. Grigorchuk, R., Savchuk, D.: Ergodic decomposition of group actions on rooted trees. In: Proc. of Steklov Inst. of Math. (to appear, 2015)

    Google Scholar 

  18. Gupta, N., Sidki, S.: On the Burnside problem for periodic groups. Math. Z. 182–3, 385–388 (1983)

    Article  MathSciNet  Google Scholar 

  19. Klimann, I.: Automaton semigroups: The two-state case. Theor. Comput. Syst., 1–17 (special issue STACS 2013) (2014)

    Google Scholar 

  20. Klimann, I., Picantin, M., Savchuk, D.: Orbit automata as a new tool to attack the order problem in automaton groups. ArXiv:1411.0158v2

  21. Muntyan, Y., Savchuk, D.: automgrp automata groups, GAP package for computation in self-similar groups and semigroups, V. 1.2.4 (2014)

    Google Scholar 

  22. Nekrashevych, V.: Self-similar groups, Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005)

    Google Scholar 

  23. Savchuk, D., Vorobets, Y.: Automata generating free products of groups of order 2. J. Algebra 336–1, 53–66 (2011)

    Article  MathSciNet  Google Scholar 

  24. Sushchansky, V.I.: Periodic permutation \(p\)-groups and the unrestricted Burnside problem. DAN SSSR. 247(3), 557–562 (1979). (in Russian)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University Paris Diderot, Sorbonne Paris Cité, LIAFA, UMR 7089 CNRS, F-75013, Paris, France

    Ines Klimann & Matthieu Picantin

  2. Department of Mathematics and Statistics, University of South Florida, 4202 E Fowler Ave, Tampa, FL, 33620-5700, USA

    Dmytro Savchuk

Authors
  1. Ines Klimann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthieu Picantin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dmytro Savchuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthieu Picantin .

Editor information

Editors and Affiliations

  1. University of Liverpool, Liverpool, United Kingdom

    Igor Potapov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Klimann, I., Picantin, M., Savchuk, D. (2015). A Connected 3-State Reversible Mealy Automaton Cannot Generate an Infinite Burnside Group. In: Potapov, I. (eds) Developments in Language Theory. DLT 2015. Lecture Notes in Computer Science(), vol 9168. Springer, Cham. https://doi.org/10.1007/978-3-319-21500-6_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21500-6_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21499-3

  • Online ISBN: 978-3-319-21500-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation