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Algorithmic Decidability of Engel’s Property for Automaton Groups

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Computer Science – Theory and Applications (CSR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9691))

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Abstract

We consider decidability problems associated with Engel’s identity (\([\cdots [[x,y],y],\dots ,y]=1\) for a long enough commutator sequence) in groups generated by an automaton.

We give a partial algorithm that decides, given x, y, whether an Engel identity is satisfied. It succeeds, importantly, in proving that Grigorchuk’s 2-group is not Engel.

We consider next the problem of recognizing Engel elements, namely elements y such that the map \(x\mapsto [x,y]\) attracts to \(\{1\}\). Although this problem seems intractable in general, we prove that it is decidable for Grigorchuk’s group: Engel elements are precisely those of order at most 2.

Our computations were implemented using the package Fr within the computer algebra system Gap.

L. Bartholdi—Partially supported by ANR grant ANR-14-ACHN-0018-01.

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References

  1. Abért, M.: Group laws and free subgroups in topological groups. Bull. Lond. Math. Soc. 37(4), 525–534 (2005). arxiv:math.GR/0306364. MR2143732

    Article  MathSciNet  MATH  Google Scholar 

  2. Akhavi, A., Klimann, I., Lombardy, S., Mairesse, J., Picantin, M.: On the finiteness problem for automaton (semi)groups. Internat. J. Algebra Comput. 22(6), 26 (2012). doi:10.1142/S021819671250052X. 1250052 MR2974106

    Article  MathSciNet  MATH  Google Scholar 

  3. Bandman, T., Grunewald, F., Kunyavskiĭ, B.: Geometry and arithmetic of verbal dynamical systems on simple groups. Groups Geom. Dyn. 4(4), 607–655 (2010). doi:10.4171/GGD/98. With an appendix by Nathan Jones. MR2727656 (2011k:14020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bartholdi, L., Grigorchuk, R.I.: On parabolic subgroups and Hecke algebras of some fractal groups. SerdicaMath. J. 28(1), 47–90 (2002). arxiv:math/9911206. MR1899368 (2003c:20027)

    MathSciNet  MATH  Google Scholar 

  5. Bartholdi, L., Grigorchuk, R.I., Šuni, Z.: Branch groups. In: Handbook of Algebra, vol. 3, pp. 989–1112. North-Holland, Amsterdam (2003). doi:10.1016/S1570-7954(03)80078-5. arxiv:math/0510294. MR2035113 (2005f:20046)

    Google Scholar 

  6. Bludov, V.V.: An example of not Engel group generated by Engel elements. In: A Conference in Honor of Adalbert Bovdi’s 70th Birthday, 18–23 November 2005, Debrecen, Hungary, pp. 7–8 (2005)

    Google Scholar 

  7. Erschler, A.: Iterated identities and iterational depth of groups (2014). arxiv:math/1409.5953

  8. The GAP Group, GAP—Groups, Algorithms, and Programming, Version 4.4.10 (2008)

    Google Scholar 

  9. Gécseg, F.: Products of Automata EATCS Monographs on Theoretical Computer Science. EATCS Monographs on Theoretical Computer Science, vol. 7. Springer, Berlin (1986). MR88b:68139b

    Google Scholar 

  10. Gécseg, F., Csákány, B.: Algebraic Theory of Automata. Akademiami Kiado, Budapest (1971)

    MATH  Google Scholar 

  11. Gillibert, P.: The finiteness problem for automaton semigroups is undecidable. Int. J. Algebra Comput. 24(1), 1–9 (2014). doi:10.1142/S0218196714500015. MR3189662

    Article  MathSciNet  MATH  Google Scholar 

  12. 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). doi:10.1007/978-3-319-15579-1_25. MR3344813

    Google Scholar 

  13. Golod, E.S.: Some problems of Burnside type. In: Proceedings of the International Congress of Mathematicians (Moscow, 1966), Izdat. “Mir”, Moscow, 1968, pp. 284–289 (Russian). MR0238880 (39 #240)

    Google Scholar 

  14. Grigorchuk, R.I.: On Burnside’s problem on periodic groups, 14(1), 53–54 (1980). English translation: Functional Anal. Appl. 14, 41–43 (1980). MR81m:20045

    Google Scholar 

  15. Grigorchuk, R.I.: On the milnor problem of group growth. Dokl. Akad. Nauk SSSR 271(1), 30–33 (1983). MR85g:20042

    MathSciNet  Google Scholar 

  16. Gruenberg, K.W.: The Engel elements of a soluble group. Illinois J. Math. 3, 151–168 (1959). MR0104730 (21 #3483)

    MathSciNet  MATH  Google Scholar 

  17. Jackson, M.: On locally finite varieties with undecidable equational theory. Algebra Univers. 47(1), 1–6 (2002). doi:10.1007/s00012-002-8169-0. MR1901727 (2003b:08002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Klimann, I., Mairesse, J., Picantin, M.: Implementing computations in automaton (Semi)groups. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 240–252. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31606-7_21. MR2993189

    Chapter  Google Scholar 

  19. Klimann, I.: The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable. In: 30th International Symposium on Theoretical Aspects of Computer Science, LIPIcs. Leibniz International Proceedings in Informatics, vol. 20, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, pp. 502–513 (2013). MR3090008

    Google Scholar 

  20. Leonov, Y.G.: On identities in groups of automorphisms of trees. Visnyk of Kyiv State University of T.G.Shevchenko 3, 37–44 (1997)

    MATH  Google Scholar 

  21. Medvedev, Y.: On compact Engel groups. Israel J. Math. 135, 147–156 (2003). doi:10.1007/BF02776054. MR1997040 (2004f:20072)

    Article  MathSciNet  MATH  Google Scholar 

  22. Neumann, H.: Varieties of Groups. Springer-Verlag New York, Inc., New York (1967). MR35#6734

    Book  MATH  Google Scholar 

  23. Straubing, H., Weil, P.: Varieties (2015). arxiv:math/1502.03951

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Acknowledgments

I am grateful to Anna Erschler for stimulating my interest in this question and for having suggested a computer approach to the problem, and to Ines Klimann and Matthieu Picantin for helpful discussions that have improved the presentation of this note.

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Authors and Affiliations

  1. École Normale Supérieure, Paris, France

    Laurent Bartholdi

  2. Georg-August-Universität zu Göttingen, Göttingen, Germany

    Laurent Bartholdi

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  1. Laurent Bartholdi
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Correspondence to Laurent Bartholdi .

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  1. Russian Academy of Sciences, St. Petersburg, Russia

    Alexander S. Kulikov

  2. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Gerhard J. Woeginger

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Bartholdi, L. (2016). Algorithmic Decidability of Engel’s Property for Automaton Groups. In: Kulikov, A., Woeginger, G. (eds) Computer Science – Theory and Applications. CSR 2016. Lecture Notes in Computer Science(), vol 9691. Springer, Cham. https://doi.org/10.1007/978-3-319-34171-2_3

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  • DOI: https://doi.org/10.1007/978-3-319-34171-2_3

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