Geometriae Dedicata

, Volume 195, Issue 1, pp 79–99 | Cite as

Growth in higher Baumslag–Solitar groups

  • Ayla P. Sánchez
  • Michael Shapiro
Original Paper


We study the HNN extension of \(\mathbb Z^m\) given by the cubing endomorphism \(g\mapsto g^3\), and prove that such groups have rational growth with respect to the standard generating sets. We compute the subgroup growth series of the horocyclic subgroup \(\mathbb Z^m\) in this family of examples, prove that for each m the subgroup has rational growth. We then use the tree-like structure of these groups to see how to compute the growth of the whole group.


Growth Baumslag–Solitar groups Solvable groups Rationality 

Mathematics Subject Classification

20F65 20F10 20F16 68Q45 



The authors would like to thank Moon Duchin, Murray Elder, and Meng-Che Ho. MS would like to thank Karen Buck for helping to jump-start some of the neurons used in this work.


  1. 1.
    Brazil, M.: Growth functions for some nonautomatic Baumslag–Solitar groups. Trans. Am. Math. Soc. 342, 137–154 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chomsky, N., Schützenberger, M.P.: The Algebraic Theory of Context-free Languages, Computer Programming and Formal Systems, pp. 118–161. North-Holland, Amsterdam (1963)CrossRefzbMATHGoogle Scholar
  3. 3.
    Ollins, D.J., Edjvet, M., Gill, C.P.: Growth series for the group \(\langle x, y | x^{-1} yx = y^l \rangle \). Arch. Math. 62(1), 1–11 (1994)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Duchin, M., Shapiro, M.: Rational growth in the Heisenberg group (
  5. 5.
    Edjvet, M., Johnson, D.L.: The growth of certain amalgamated free products and HNN extensions. J. Aust. Math. Soc. Ser. A 52(3), 285–298 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Freden, E.M.: Chapter 12: growth of groups. In: Clay, M., Margalit, D. (eds.) Office Hours with a Geometric Group Theorist. Princeton University Press, Princeton (2017)Google Scholar
  7. 7.
    Freden, E.M., Knudson, T., Schofield, J.: Growth in Baumslag–Solitar groups I: subgroups and rationality. LMS J. Comput. Math. 14, 34–71 (2011)Google Scholar
  8. 8.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages and Computability, 2nd edn. Addison-Wesley Longman, Boston (2000)zbMATHGoogle Scholar
  9. 9.
    Mann, A.: How groups grow London mathematical society lecture note series, 395. Cambridge University Press, Cambridge (2012)Google Scholar
  10. 10.
    Shapiro, M.: Growth of a \(PSL_2{\mathbb{R}}\) manifold group. Math. Nachr. 167, 279–312 (1994)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Stoll, M.: Rational and transcendental growth series for the higher Heisenberg groups. Invent. Math. 126, 85–109 (1996)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Wheaton CollegeNortonUnited States
  2. 2.University of BathBathUK

Personalised recommendations