Abstract
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.
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Notes
An anonymous referee pointed out to MS that the obvious proof here requires the group to be recursively presented. This leaves open the question as to whether there is a group with unsolvable word problem and rational growth. An answer either way would be fascinating.
References
Brazil, M.: Growth functions for some nonautomatic Baumslag–Solitar groups. Trans. Am. Math. Soc. 342, 137–154 (1994)
Chomsky, N., Schützenberger, M.P.: The Algebraic Theory of Context-free Languages, Computer Programming and Formal Systems, pp. 118–161. North-Holland, Amsterdam (1963)
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)
Duchin, M., Shapiro, M.: Rational growth in the Heisenberg group (http://arxiv.org/abs/1411.4201)
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)
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)
Freden, E.M., Knudson, T., Schofield, J.: Growth in Baumslag–Solitar groups I: subgroups and rationality. LMS J. Comput. Math. 14, 34–71 (2011)
Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages and Computability, 2nd edn. Addison-Wesley Longman, Boston (2000)
Mann, A.: How groups grow London mathematical society lecture note series, 395. Cambridge University Press, Cambridge (2012)
Shapiro, M.: Growth of a \(PSL_2{\mathbb{R}}\) manifold group. Math. Nachr. 167, 279–312 (1994)
Stoll, M.: Rational and transcendental growth series for the higher Heisenberg groups. Invent. Math. 126, 85–109 (1996)
Acknowledgements
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.
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Sánchez, A.P., Shapiro, M. Growth in higher Baumslag–Solitar groups. Geom Dedicata 195, 79–99 (2018). https://doi.org/10.1007/s10711-017-0277-2
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DOI: https://doi.org/10.1007/s10711-017-0277-2