Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-19T00:34:16.220Z Has data issue: false hasContentIssue false
  • English
  • Français

The Dehn function and a regular set of normal forms for R. Thompson's group F

Part of: Low-dimensional topology Connections with homological algebra and category theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

V. S. Guba  and
M. V. Sapir
V. S. Guba
Affiliation:
Department of Mathematics Vologda State Pedagogical InstituteS. Orlov Street 6 160600, VologdaRussia e-mail: [email protected]
M. V. Sapir
Affiliation:
Department of Mathematics and Statistics University of Nebraska-LincolnCenter for Communication and Information Science Lincoln, NE 68588-0323USA 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 show that the group F discovered by Richard Thompson in 1965 has a subexponential upper bound for its Dehn function. This disproves a conjecture by Gersten. We also prove that F has a regular terminating confluent presentation.

MSC classification

Secondary: 20J05: Homological methods in group theory 20F06: Cancellation theory; application of van Kampen diagrams 20F10: Word problems, other decision problems, connections with logic and automata 57M07: Topological methods in group theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 3 , June 1997 , pp. 315 - 328
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Baumslag, G., Miller, C. F. and Short, H., ‘Isoperimetric inequalities and the homology of groups’, Invent. Math. 113 (1993), 531–560.CrossRefGoogle Scholar
[2]Brin, M. G. and Squier, C. C., ‘Groups of piecewise homeomorphisms of the real line’, Invent. Math. 79 (1985), 485–498.CrossRefGoogle Scholar
[3]Brown, K. S., ‘Finiteness properties of groups’, J. Pure Appl. Algebra 44 (1987), 45–75.CrossRefGoogle Scholar
[4]Brown, K. S. and Geoghegan, R., ‘An infinite-dimensional torsion-free fp∞ group’, Invent. Math. 77 (1984), 367–381.Google Scholar
[5]Cannon, J. W., Floyd, W. J. and Parry, W. R., ‘Notes on richard thompson's groups f and t’, Technical report, (Geometry Center, University of Minnesota, 1994).Google Scholar
[6]Cohen, D., ‘String rewriting – a survey for group theorists’, in: Geometric group theory (Cambridge Univ. Press, Cambridge, 1993) pp. 37–47. (Sussex, 1991) 1.CrossRefGoogle Scholar
[7]Dershowitz, N. and Jouannaud, J.-P., ‘Rewrite systems’, in: Handbook of theoretical computer science (ed. van Leeuwen, J.) (Elsevier Science Publishers, Amsterdam, 1990) chapter 6, pp. 244–320.Google Scholar
[8]Eilenberg, S., Automata, languages and machines, volume B (Academic Press, New York, 1976).Google Scholar
[9]Gersten, S., “Thompson's group f is not combable”, preprint, University of Utah.Google Scholar
[10]Gersten, S., ‘Isoperimetric and isodiametric functions of finite presentations’, in: Geometric group theory (Cambridge Univ. Press, Cambridge, 1993) pp. 79–96. (Sussex, 1991) 1.CrossRefGoogle Scholar
[11]Gromov, M., ‘Hyperbolic groups’, in: Essays in group theory (ed. Gersten, S.), Math. Sci. Res. Inst. Publ 8 (Springer, Berlin, 1987) pp. 75–263.CrossRefGoogle Scholar
[12]Guba, V. S. and Sapir, M. V., ‘Diagram groups’, Trans. Amer. Math. Soc. to appear.Google Scholar
[13]Kilibarda, V., On the algebra of semigroup diagrams (Ph.D. Thesis, University of Nebraska, 1994).Google Scholar
[14]Madlener, K. and Otto, F., ‘Pseudo-natural algorithms for the word problem for finitely presented monoids and groups’, J. Symbolic Comp. 1 (1985), 383–418.CrossRefGoogle Scholar
[15]McKenzie, R. and Thompson, R. J., ‘An elementary construction of unsolvable word problem in group theory’, in: Word problems (eds. Boon, W. W., Cannonito, F. B. and Lyndon, R. C.) (North-Holland, Amsterdam, 1973) pp. 457–478.Google Scholar
[16]Newman, M. H. A., ‘On theories with a combinatorial definition of equivalence’, Ann. Math. 43 (1942), 223–243.CrossRefGoogle Scholar
[17]Ol'shanskii, A. Yu., ‘Hyperbolicity of groups with subquadratic isoperimetric inequality’, Internat. J. Algebra Comp. 1 (1991), 281–290.CrossRefGoogle Scholar
[18]Squier, C. C., ‘Word problems and a homological finiteness condition for monoids’, J. Pure Appl. Algebra 49 (1987), 201–217.CrossRefGoogle Scholar