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Connected Reversible Mealy Automata of Prime Size Cannot Generate Infinite Burnside Groups

Authors Thibault Godin, Ines Klimann

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Keywords
  • Burnside problem
  • automaton groups
  • reversibility
  • orbit trees

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Abstract

The simplest example of an infinite Burnside group arises in the class of automaton groups. However there is no known example of such a group generated by a reversible Mealy automaton. It has been proved that, for a connected automaton of size at most 3, or when the automaton is not bireversible, the generated group cannot be Burnside infinite. In this paper, we extend these results to  automata with bigger stateset, proving that, if a connected reversible automaton  has a prime number of states, it cannot generate an infinite Burnside group.

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Thibault Godin and Ines Klimann. Connected Reversible Mealy Automata of Prime Size Cannot Generate Infinite Burnside Groups. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 44:1-44:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.MFCS.2016.44

Author Details

Thibault Godin
Ines Klimann

References

  1. A. Akhavi, I. Klimann, S. Lombardy, J. Mairesse, and M. Picantin. On the finiteness problem for automaton (semi)groups. Int. J. Algebr. Comput., 22(6):26p., 2012. Google Scholar
  2. S.V. Alešin. Finite automata and the Burnside problem for periodic groups. Mat. Zametki, 11:319-328, 1972. Google Scholar
  3. W. Burnside. On an unsettled question in the theory of discontinuous groups. Quart. J. Math., 33:230-238, 1902. Google Scholar
  4. D. D'Angeli and E. Rodaro. A geometric approach to (semi)-groups defined by automata via dual transducers. Geometriae Dedicata, 174-1:375-400, 2015. Google Scholar
  5. P.W. Gawron, V.V. Nekrashevych, and V.I. Sushchansky. Conjugation in tree automorphism groups. Internat. J. Algebr. Comput., 11-5:529-547, 2001. Google Scholar
  6. V.M. Gluškov. Abstract theory of automata. Uspehi Mat. Nauk, 16-5:3-62, 1961. Google Scholar
  7. Th. Godin, I. Klimann, and M. Picantin. On torsion-free semigroups generated by invertible reversible Mealy automata. In LATA'15, volume 8977 of LNCS, pages 328-339, 2015. Google Scholar
  8. E.S. Golod. On nil-algebras and finitely residual groups. Izv. Akad. Nauk SSSR. Ser. Mat., 28:273-276, 1964. Google Scholar
  9. E.S. Golod and I. Shafarevich. On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat., 28:261-272, 1964. Google Scholar
  10. R. Grigorchuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14-1:53-54, 1980. Google Scholar
  11. R. Grigorchuk and D. Savchuk. Ergodic decomposition of group actions on rooted trees. to appear in Proc. of Steklov Inst. of Math., 2015. Google Scholar
  12. I. Klimann. Automaton semigroups: The two-state case. Theor. Comput. Syst. (special issue STACS'13), pages 1-17, 2014. URL: http://dx.doi.org/10.1007/s00224-014-9594-0.
  13. I. Klimann, M. Picantin, and D. Savchuk. A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group. In DLT' 15, volume 9168 of LNCS, pages 313-325, 2015. Google Scholar
  14. E.I. Zel´manov Solution of the restricted Burnside problem for groups of odd exponent. Izv. AN SSSR Math+, 54-1:42-59, 221, 1990. Google Scholar
  15. E.I. Zel´manov. Solution of the restricted Burnside problem for 2-groups. Mat. Sb., 182-4:568-592, 1991. Google Scholar
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