Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T15:12:07.540Z Has data issue: false hasContentIssue false
  • English
  • Français

GROUPS AND SEMIGROUPS WITH A ONE-COUNTER WORD PROBLEM

Part of: Computability and recursion theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 October 2008

DEREK F. HOLT ,
MATTHEW D. OWENS  and
RICHARD M. THOMAS
DEREK F. HOLT*
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: [email protected])
MATTHEW D. OWENS
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
RICHARD M. THOMAS
Affiliation:
Department of Computer Science, University of Leicester, Leicester LE1 7RH, UK (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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.

We prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.

Keywords

groupssemigroupsword problem

MSC classification

Secondary: 20F10: Word problems, other decision problems, connections with logic and automata 03D40: Word problems, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 2 , October 2008 , pp. 197 - 209
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Anisimov, A. V., ‘Certain algorithmic problems for groups and context-free languages’, Kibernetica 2 (1972), 4–11.Google Scholar
[2]Berstel, Jean, Transductions and Context-Free Languages (Teubner, Stuttgart, 1979).CrossRefGoogle Scholar
[3]Duncan, Andrew and Gilman, Robert H., ‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc. 136 (2004), 513–524.Google Scholar
[4]Dunwoody, M. J., ‘The accessibility of finitely presented groups’, Invent. Math. 81 (1985), 449–457.Google Scholar
[5]Elder, Murray, Kambites, Mark and Ostheimer, Gretchen, On groups and counter automata, arXiv:math/0611188v1, 7 Nov 2006.Google Scholar
[6]Epstein, David B. A., Cannon, J. W., Holt, D. F., Levy, S., Paterson, M. S. and Thurston, W. P., Word Processing in Groups (Jones and Bartlett, Boston, MA, 1992).CrossRefGoogle Scholar
[7]Gromov, Mikhael, ‘Groups of polynomial growth and expanding maps’, Publ. Math. Inst. Hautes Études Sci. 53 (1981), 53–78.Google Scholar
[8]Herbst, Thomas, ‘On a subclass of context-free groups’, RAIRO Inform. Théor. Appl. 25 (1991), 255–272.Google Scholar
[9]Herbst, Thomas and Thomas, Richard M., ‘Group presentations, formal languages and characterizations of one-counter groups’, Theoret. Comput. Sci. 112 (1993), 187–213.Google Scholar
[10]Holt, Derek F., Röver, Claas E., Rees, Sarah and Thomas, Richard M., ‘Groups with a context-free co-word problem’, J. London Math. Soc. 71 (2005), 643–657.CrossRefGoogle Scholar
[11]Justin, J., ‘Groupes et semi-groupes à croissance linéaire’, C. R. Acad. Sci. Paris Sér. A–B 273 (1971), 212–214.Google Scholar
[12]Muller, David E. and Schupp, Paul E., ‘Groups, the theory of ends, and context-free languages’, J. Comput. System Sci. 26 (1983), 295–310.Google Scholar
[13]Neumann, B. H., ‘Groups covered by permutable subsets’, J. London Math. Soc. 29 (1954), 236–248.Google Scholar
[14]Parikh, Rohit J., ‘Language-generating devices’, Quarterly Research Reports, 60, Research Laboratory of Electronics, MIT, Cambridge, MA, 1961, pp. 199–212.Google Scholar
[15]Stallings, John, Group Theory and Three-Dimensional Manifolds, Yale Mathematical Monographs, 4 (Yale University Press, New Haven, CT and London, 1971).Google Scholar
[16]Wilkie, A. J. and van den Dries, L., ‘An effective bound for groups of linear growth’, Arch. Math. (Basel) 42 (1984), 391–396.Google Scholar