Measuring Closeness Between Cayley Automatic Groups and Automatic Groups

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10792)


In this paper we introduce a way to estimate a level of closeness of Cayley automatic groups to the class of automatic groups using a certain numerical characteristic. We characterize Cayley automatic groups which are not automatic in terms of this numerical characteristic and then study it for the lamplighter group, the Baumslag–Solitar groups and the Heisenberg group.


Automatic groups Cayley automatic groups Automatic structures Numerical characteristics of groups Lamplighter group Heisenberg group Baumslag–Solitar groups 



The authors thank the referees for useful comments.


  1. 1.
    Baumslag, G., Solitar, D.: Some two-generator one-relator non-Hopfian groups. Bull. Am. Math. Soc. 68, 199–201 (1962)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berdinsky, D., Khoussainov, B.: On automatic transitive graphs. In: Shur, A.M., Volkov, M.V. (eds.) DLT 2014. LNCS, vol. 8633, pp. 1–12. Springer, Cham (2014). Google Scholar
  3. 3.
    Berdinsky, D., Khoussainov, B.: Cayley automatic representations of wreath products. Int. J. Found. Comput. Sci. 27(2), 147–159 (2016)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blachere, S.: Word distance on the discrete Heisenberg group. Colloq. Math. 95(1), 21–36 (2003)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Burillo, J., Elder, M.: Metric properties of Baumslag-Solitar groups. Int. J. Algebra Comput. 25(5), 799–811 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cleary, S., Taback, J.: Dead end words in lamplighter groups and other wreath products. Q. J. Math. 56(2), 165–178 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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 Barlett Publishers, Boston (1992)MATHGoogle Scholar
  8. 8.
    Kargapolov, M.I., Merzljakov, J.I.: Fundamentals of the Theory of Groups. Springer, Heidelberg (1979)Google Scholar
  9. 9.
    Kharlampovich, O., Khoussainov, B., Miasnikov, A.: From automatic structures to automatic groups. Groups, Geom. Dyn. 8(1), 157–198 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995). CrossRefGoogle Scholar
  11. 11.
    Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Heidelberg (1977).
  12. 12.
    Roe, J.: Lectures on Coarse Geometry. University Lecture Series, vol. 31. American Mathematical Society, Providence (2003)Google Scholar
  13. 13.
    Vershik, A.: Numerical characteristics of groups and corresponding relations. J. Math. Sci. 107(5), 4147–4156 (2001)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMahidol UniversityBangkokThailand
  2. 2.Centre of Excellence in MathematicsCommission on Higher EducationBangkokThailand

Personalised recommendations