Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Growth functions for some nonautomatic Baumslag-Solitar groups


Author: Marcus Brazil
Journal: Trans. Amer. Math. Soc. 342 (1994), 137-154
MSC: Primary 20F10
DOI: https://doi.org/10.1090/S0002-9947-1994-1169911-8
MathSciNet review: 1169911
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The growth function of a group is a generating function whose coefficients $ {a_n}$ are the number of elements in the group whose minimum length as a word in the generators is n. In this paper we use finite state automata to investigate the growth function for the Baumslag-Solitar group of the form $ \langle a,b\vert{a^{ - 1}}ba = {a^2}\rangle $ based on an analysis of its combinatorial and geometric structure. In particular, we obtain a set of length-minimal normal forms for the group which, although it does not form the language of a finite state automata, is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational. The rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic.

The results in this paper generalize to the groups $ \langle a,b\vert{a^{ - 1}}ba = {a^m}\rangle $ for all positive integers m.


References [Enhancements On Off] (What's this?)

  • [1] G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199-201. MR 0142635 (26:204)
  • [2] G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, MSRI preprint. MR 1147304 (93a:20048)
  • [3] M. Brazil, Monoid growth functions for braid groups, Internat. J. Algebra and Comput. 1 (1991), 201-205. MR 1128012 (92h:20063)
  • [4] -Groups with rational growth, Ph.D. Thesis, La Trobe University, 1992.
  • [5] M. Edjvet and D. L. Johnson, The growth of certain amalgamated free products and HNN-extensions, J. Austral. Math. Soc. (to appear). MR 1151287 (93b:20048)
  • [6] D. B. A. Epstein, J. W. Cannon, D. F. Holt, M. S. Patterson and W. P. Thurston, Word processing in groups, Jones and Bartlett, Boston, Mass., 1992. MR 1161694 (93i:20036)
  • [7] E. Ghyst and P. de la Harpe (eds.), Sur les groupes hyperboliques d'apres Mikhael Gromov, Birkhäuser, Boston, Mass., 1990. MR 1086648 (92f:53050)
  • [8] U. Zwick, Computing growth functions, University of Warwick, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20F10

Retrieve articles in all journals with MSC: 20F10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1169911-8
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society