Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T09:01:38.374Z Has data issue: false hasContentIssue false
  • English
  • Français

The growth of certain amalgamated free products and HNN-extensions

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

M. Edjvet  and
D. L. Johnson
M. Edjvet
Affiliation:
University of NottinghamUniversity Park Nottingham NG7 2RDU.K.
D. L. Johnson
Affiliation:
University of NottinghamUniversity Park Nottingham NG7 2RDU.K.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Here we mean growth in the sense of Milnor and Gromov. After a brief survey of known results, we compute the growth series of the groups , with respect to generators {x, y}. This is done using minimal normal forms obtained by informal use of judiciously chosen rewrite rules. In both of these examples the growth series is a rational function, and we suspect that this is not the case for the Baumslag-Solitar group

.

MSC classification

Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 3 , June 1992 , pp. 285 - 298
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Benson, M., ‘On the rational growth of virtually nilpotent groups’, Combinatorial Group Theory and Topology, Princeton Studies no. 111, pp. 185–196 (Princeton University Press, Princeton 1987).CrossRefGoogle Scholar
[2]Billington, N., The growth of stratified groups (Ph D thesis, La Trobe University, 1986).Google Scholar
[3]Cannon, J. W., ‘The growth of the closed surface groups and compact hyperbolic groups’, preprint.Google Scholar
[4]Govorov, V. E., ‘On the dimension of graded algebras’, Mat Zametki 14 (1973), 209–216Google Scholar
(translation: Math. Notes 14 (1973), 678–682).Google Scholar
[5]Grigorchuk, R. I., ‘On Milnor's problem of group growth,’ Dokl. Akad. Nauk SSSR 271: 1 (1983)Google Scholar
(translation: Soviet Math. Dokl. 28 (1983), 23–26).Google Scholar
[6]Groves, J. R. J. and Smith, G. C., ‘Rewriting systems and soluble groups’, Bath Comp Sci Tech Report no. 19 (1989).Google Scholar
[7]Grünewald, F., seminar at Berkeley, 01 1989.Google Scholar
[8]Gupta, N. D. and Sidki, S., ‘On the Burnside problem for periodic groups’, Math. Z. 182 (1983), 385–388.CrossRefGoogle Scholar
[9]Johnson, D. L., ‘Rational growth of wreath products,’ in Groups St. Andrews 1989 (ed. Campbell, C. M. and Robertson, E. F.), pp. 309–315, Cambridge Univ. Press, 1991.Google Scholar
[10]Le Chedance, P., Canonical forms in finitely presented algebras (Pitman, London 1986).Google Scholar
[11]Lorenz, M., Gel1 fand-Kirillov dimension and Poincaré series (Lecture Notes, De Kalb 1987).Google Scholar
[12]Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory (Springer, Berlin 1977).Google Scholar
[13]Magnus, W., Noneuclidean tessellations and their groups (Academic Press, New York-London, 1974).Google Scholar
[14]Milnor, J., ‘A note on curvature and the fundamental group’, J. Differential Geom. 2 (1968), 1–7.CrossRefGoogle Scholar
[15]Smythe, N. J., ‘Growth functions and Euler series,’ Invent. Math. 77 (1984), 517–531.CrossRefGoogle Scholar