To Infinity and Beyond

Author Ines Klimann



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Author Details

Ines Klimann
  • Univ Paris Diderot, Sorbonne Paris Cité, IRIF, UMR 8243 CNRS, F-75013 Paris, France

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Ines Klimann. To Infinity and Beyond. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 131:1-131:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.131

Abstract

We prove that if a group generated by a bireversible Mealy automaton contains an element of infinite order, then it must have exponential growth. As a direct consequence, no infinite virtually nilpotent group can be generated by a bireversible Mealy automaton.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Automata over infinite objects
Keywords
  • automaton groups
  • growth of a group
  • exponential growth

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References

  1. L. Bartholdi and A. Erschler. Groups of given intermediate word growth. Ann. Inst. Fourier, 64(5):2003-2036, 2014. Google Scholar
  2. T. Brough and A.J. Cain. Automaton semigroups: New constructions results and examples of non-automaton semigroups. Theoretical Computer Science, 674:1-15, 2017. Google Scholar
  3. C. Drutu and M. Kapovich. Lectures on geometric group theory. URL: https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf.
  4. Th. Godin and I. Klimann. On bireversible Mealy automata and the Burnside problem. Theoretical Computer Science, 707:24-35, 2018. Google Scholar
  5. R. Grigorchuk. Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat., 48-5:939-985, 1984. Google Scholar
  6. R. Grigorchuk. Degrees of growth of finitely generated groups, and the theory of invariant means. Mathematics of the USSR-Izvestiya, 25(2):259, 1985. Google Scholar
  7. R. Grigorchuk and I. Pak. Groups of intermediate growth: an introduction. L'Enseignement Mathématique, 54(3-4):251-272, 2008. Google Scholar
  8. R.I. Grigorchuk. On periodic groups generated by finite automata. In 18-th USSR Algebraic Conference, Kishinev, 1985. Google Scholar
  9. I. Klimann. Automaton semigroups: The two-state case. Theor. Comput. Syst. (special issue STACS'13), pages 1-17, 2014. Google Scholar
  10. I. Klimann. On level-transitivity and exponential growth. Semigroup Forum, 95(3):441-447, 2016. Google Scholar
  11. 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
  12. A. Mann. How groups grow, volume 395 of Lecture Note Series. The London Mathematical Society, 2012. Google Scholar
  13. J. Milnor. Advanced problem 5603. MAA Monthly, 75:685-686, 1968. Google Scholar
  14. V. Nekrashevich. Palindromic subshifts and simple periodic groups of intermediate growth. Annals of Mathematics, 187(3):667-719, 2018. Google Scholar
  15. F. Olukoya. The growth rates of automaton groups generated by reset automata. arXiv:1708.07209, 2017. Google Scholar
  16. J.A. Wolf. Growth of finitely generated solvable groups and curvature of riemanniann manifolds. J. Differential Geom., 2(4):421-446, 1968. Google Scholar
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