Automaton (Semi)groups: Wang Tilings and Schreier Tries

  • Ines Klimann
  • Matthieu Picantin
Part of the Trends in Mathematics book series (TM)


Groups and semigroups generated by Mealy automata were formally introduced in the early 1960s. They revealed their full potential over the years, by contributing to important conjectures in group theory. In the current chapter, we intend to provide various combinatorial and dynamical tools to tackle some decision problems all related to some extent to the growth of automaton (semi)groups. In the first part, we consider Wang tilings as a major tool in order to study and understand the behavior of automaton (semi)groups. There are various ways to associate a Wang tileset with a given complete and deterministic Mealy automaton and various ways to interpret the induced Wang tilings. We describe some of these fruitful combinations, as well as some promising research opportunities. In the second part, we detail some toggle switch between a classical notion from group theory—Schreier graphs—and some properties of an automaton group about its growth or the growth of its monogenic subgroups. We focus on polynomial-activity automata and on reversible automata, which are somehow diametrically opposed families.


  1. 9.
    Akhavi, A., Klimann, I., Lombardy, S., Mairesse, J., Picantin, M.: On the finiteness problem for automaton (semi)groups. Int. J. Algebra Comput. 22(6), 1–26 (2012)MathSciNetCrossRefGoogle Scholar
  2. 11.
    Alešin, S.: Finite automata and the Burnside problem for periodic groups. Mat. Zametki 11, 319–328 (1972)Google Scholar
  3. 42.
    Bartholdi, L., Grigorchuk, R.I., Šuniḱ, Z.: Handbook of Algebra, vol. 3, chap. Branch groups, pp. 989–1112. Elsevier BV (2003)Google Scholar
  4. 72.
    Berger, R.: The undecidability of the domino problem. Mem. Am. Math. Soc. 66, 72 (1966)Google Scholar
  5. 97.
    Bondarenko, I., Bondarenko, N., Sidki, S., Zapata, F.: On the conjugacy problem for finite-state automorphisms of regular rooted trees. Groups Geom. Dyn. 7(2), 323–355 (2013)MathSciNetCrossRefGoogle Scholar
  6. 98.
    Bondarenko, I.V.: Growth of Schreier graphs of automaton groups. Math. Ann. 354(2), 765–785 (2012)MathSciNetCrossRefGoogle Scholar
  7. 108.
    Brough, T., Cain, A.J.: Automaton semigroup constructions. Semigroup Forum 90(3), 763–774 (2015)MathSciNetCrossRefGoogle Scholar
  8. 109.
    Brough, T., Cain, A.J.: Automaton semigroups: new constructions results and examples of non-automaton semigroups. Theor. Comput. Sci. 674, 1–15 (2017)MathSciNetCrossRefGoogle Scholar
  9. 122.
    Burnside, W.: On an unsettled question in the theory of discontinuous groups. Q. J. Math. 33, 230–238 (1902)Google Scholar
  10. 123.
    Cain, A.: Automaton semigroups. Theor. Comput. Sci. 410, 5022–5038 (2009)MathSciNetCrossRefGoogle Scholar
  11. 170.
    Culik II, K.: An aperiodic set of 13 Wang tiles. Discret. Math. 160, 245–251 (1996)MathSciNetCrossRefGoogle Scholar
  12. 171.
    Culik II, K., Pachl, J.K., Yu, S.: On the limit sets of cellular automata. SIAM J. Comput. 18(4), 831–842 (1989)MathSciNetCrossRefGoogle Scholar
  13. 177.
    D’Angeli, D., Godin, T., Klimann, I., Picantin, M., Rodaro, E.: Boundary action of automaton groups without singular points and Wang tilings (2016). ArXiv:1604.07736Google Scholar
  14. 178.
    D’Angeli, D., Rodaro, E.: A geometric approach to (semi)-groups defined by automata via dual transducers. Geom. Dedicata 174-1, 375–400 (2015)MathSciNetCrossRefGoogle Scholar
  15. 184.
    Dehornoy, P.: Garside and quadratic normalisation: a survey. In: 19th International Conference on Developments in Language Theory (DLT 2015). Lecture Notes in Computer Science, vol. 9168, pp. 14–45 (2015)CrossRefGoogle Scholar
  16. 185.
    Dehornoy, P., Guiraud, Y.: Quadratic normalization in monoids. Int. J. Algebra Comput. 26(5), 935–972 (2016)MathSciNetCrossRefGoogle Scholar
  17. 186.
    Dehornoy, P., et al.: Foundations of Garside theory. Eur. Math. Soc. Tracts in Mathematics, vol. 22 (2015) garside/Garside.pdfGoogle Scholar
  18. 190.
    Delacourt, M., Ollinger, N.: Permutive one-way cellular automata and the finiteness problem for automaton groups. In: 13th Conference on Computability in Europe (CiE 2017). Lecture Notes in Computer Science, vol. 10307, pp. 234–245 (2017)CrossRefGoogle Scholar
  19. 215.
    Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Bartlett Publishers, Boston, MA (1992)Google Scholar
  20. 238.
    Gawron, P.W., Nekrashevych, V.V., Sushchansky, V.I.: Conjugation in tree automorphism groups. Int. J. Algebra Comput. 11(5), 529–547 (2001)MathSciNetCrossRefGoogle Scholar
  21. 242.
    Gillibert, P.: The finiteness problem for automaton semigroups is undecidable. Int. J. Algebra Comput. 24(1), 1–9 (2014)MathSciNetCrossRefGoogle Scholar
  22. 243.
    Gillibert, P.: Simulating Turing machines with invertible Mealy automata (2017, in preparation)Google Scholar
  23. 246.
    Glasner, Y., Mozes, S.: Automata and square complexes. Geom. Dedicata 111, 43–64 (2005)MathSciNetCrossRefGoogle Scholar
  24. 247.
    Gluškov, V.: Abstract theory of automata. Uspehi Mat. Nauk 16(5), 3–62 (1961)Google Scholar
  25. 251.
    Godin, T., Klimann, I.: Connected reversible Mealy automata of prime size cannot generate infinite Burnside groups. In: 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). LIPIcs, vol. 58, pp. 44:1–44:14 (2016)Google Scholar
  26. 252.
    Godin, T., Klimann, I., Picantin, M.: On torsion-free semigroups generated by invertible reversible Mealy automata. In: 9th International Conference on Language and Automata Theory and Applications (LATA). Lecture Notes in Computer Science, vol. 8977, pp. 328–339 (2015)zbMATHGoogle Scholar
  27. 255.
    Golod, E.S.: On nil-algebras and finitely residual groups. Izv. Akad. Nauk SSSR. Ser. Mat. 28, 273–276 (1964)Google Scholar
  28. 256.
    Golod, E.S., Shafarevich, I.: On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28, 261–272 (1964)Google Scholar
  29. 266.
    Grigorchuk, R.I.: On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1), 53–54 (1980)Google Scholar
  30. 269.
    Grigorchuk, R.I.: 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
  31. 270.
    Grigorchuk, R.I.: New Horizons in pro-p Groups, chap. Just Infinite Branch Groups, pp. 121–179. Birkhäuser, Boston, MA (2000)CrossRefGoogle Scholar
  32. 271.
    Grigorchuk, R.I., Nekrashevich, V.V., Sushchanskiı̆, V.I.: Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231, 134–214 (2000)Google Scholar
  33. 272.
    Grigorchuk, R.I., Nekrashevych, V.V.: Amenable actions of non-amenable groups. J. Math. Sci. 140(3), 391–397 (2007)MathSciNetCrossRefGoogle Scholar
  34. 296.
    Hoffmann, M.: Automatic semigroups. Ph.D. thesis, Univ Leicester (2001)Google Scholar
  35. 311.
    Jeandel, E., Rao, M.: An aperiodic set of 11 Wang tiles (2015). ArXiv:1506.06492Google Scholar
  36. 332.
    Kari, J.: The nilpotency problem of one-dimensional cellular automata. SIAM J. Comput. 21(3), 571–586 (1992)MathSciNetCrossRefGoogle Scholar
  37. 333.
    Kari, J.: A small aperiodic set of Wang tiles. Discret. Math. 160, 259–264 (1996)MathSciNetCrossRefGoogle Scholar
  38. 336.
    Kari, J., Ollinger, N.: Periodicity and immortality in reversible computing. In: 33rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2008). LNCS, vol. 5162, pp. 419–430 (2008)Google Scholar
  39. 351.
    Klimann, I.: Automaton semigroups: the two-state case. Theory Comput. Syst. 1–17 (2014)Google Scholar
  40. 352.
    Klimann, I.: On level-transitivity and exponential growth. Semigroup Forum, pp. 1–7 (2016)MathSciNetCrossRefGoogle Scholar
  41. 353.
    Klimann, I., Picantin, M., Savchuk, D.: A connected 3-state reversible Mealy automaton cannot generate an infinite burnside group. In: 19th International Conference on Developments in Language Theory (DLT). Lecture Notes in Computer Science, vol. 9168, pp. 313–325 (2015)CrossRefGoogle Scholar
  42. 354.
    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)MathSciNetCrossRefGoogle Scholar
  43. 370.
    Le Gloannec, B.: The 4-way deterministic periodic domino problem is undecidable. HAL:00985482 (2014)Google Scholar
  44. 409.
    Mann, A.: How Groups Grow. Lecture Note Series, vol. 395. London Mathematical Society (2012)Google Scholar
  45. 419.
    Milnor, J.: Problem 5603. Am. Math. Mon. 75(6), 685–686 (1968)Google Scholar
  46. 441.
    Nekrashevych, V.V.: Self-Similar Groups. Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence, RI (2005)Google Scholar
  47. 482.
    Picantin, M.: Automatic semigroups vs automaton semigroups (2016). ArXiv:1609.09364Google Scholar
  48. 483.
    Picantin, M.: Automates, (semi)groupes, dualités. Habilitation à diriger des recherches, Univ. Paris Diderot (2017)Google Scholar
  49. 521.
    Savchuk, D., Vorobets, Y.: Automata generating free products of groups of order 2. J. Algebra 336-1(1), 53–66 (2011)MathSciNetCrossRefGoogle Scholar
  50. 537.
    Serre, J.P.: Trees. Springer, Berlin (1980)CrossRefGoogle Scholar
  51. 546.
    Sidki, S.: Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. 100(1), 1925–1943 (2000)MathSciNetCrossRefGoogle Scholar
  52. 549.
    Silva, P.V., Steinberg, B.: On a class of automata groups generalizing lamplighter groups. Int. J. Algebra Comput. 15(5-6), 1213–1234 (2005)MathSciNetCrossRefGoogle Scholar
  53. 556.
    Šuniḱ, Z., Ventura, E.: The conjugacy problem in automaton groups is not solvable. J. Algebra 364(0), 148–154 (2012)Google Scholar
  54. 574.
    Vaughan-Lee, M.: The restricted Burnside problem. London Mathematical Society Monographs. New Series, vol. 8. Oxford University Press, Oxford (1993)Google Scholar
  55. 580.
    Wang, H.: Proving theorems by pattern recognition, II. Bell Syst. Tech. J. 40(1), 1–41 (1961)CrossRefGoogle Scholar
  56. 593.
    Zelmanov, E.I.: Solution of the restricted Burnside problem for groups of odd exponent. Izv. Akad. Nauk SSSR Ser. Mat. 54(1), 42–59, 221 (1990)MathSciNetCrossRefGoogle Scholar
  57. 594.
    Zelmanov, E.I.: Solution of the restricted Burnside problem for 2-groups. Matematicheskiı̆ Sbornik 182(4), 568–592 (1991)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.IRIFUMR 8243, CNRS & Université Paris Diderot - Case 7014Paris Cedex 13France

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