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A bicombing that implies a sub-exponential isoperimetric inequality

Published online by Cambridge University Press:  20 January 2009

Günther Huck  and
Stephan Rosebrock
Günther Huck
Affiliation:
Institut F. Didaktik der Mathematik, J.-W.-Goethe Universität, Senckenberganlage 9, 6000 Frankfurt/M., West Germany
Stephan Rosebrock
Affiliation:
Department of Mathematics, Northern Arizona University, Flagstaff AZ 86011, USA
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Abstract

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The idea of applying isoperimetric functions to group theory is due to M. Gromov [8]. We introduce the concept of a “bicombing of narrow shape” which generalizes the usual notion of bicombing as defined for example in [5], [2], and [10]. Our bicombing is related to but different from the combings defined by M. Bridson [4]. If they Cayley graph of a group with respect to a given set of generators admits a bicombing of narrow shape then the group is finitely presented and satisfies a sub-exponential isoperimetric inequality, as well as a polynomial isodiametric inequality. We give an infinite class of examples which are not bicombable in the usual sense but admit bicombings of narrow shape.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 3 , October 1993 , pp. 515 - 523
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Alonso, J. M., Combings of groups, in: Algorithms and Classification in Combinatorial Group Theory, Baumslag, G. and Miller, C. F. III, eds. (Springer Verlag, MSRI Publ., 1991).Google Scholar
2.Alonso, J. M. and Bridson, M. R., Semihyperbolic groups (preprint, Cornell University, 1990).Google Scholar
3.Bogley, W., Unions of Cockcroft two-complexes, preprint, 1991.Google Scholar
4.Bridson, M., On the geometry of normal forms in discrete groups (preprint, Princeton University, 1992).Google Scholar
5.Epstein, David B. H., Cannon, J. W., Holt, D. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P., Word processing in groups (Jones and Bartlett, 1992).Google Scholar
6.Gersten, S., Dehn functions and l 1-norms of finite presentations, in: Proceedings of the Workshop on Algorithmic Problems, Miller, C. F. III and Baumslag, G., eds. (Springer Verlag, 1991).Google Scholar
7.Gersten, S., Isoperimetric and isodiametric functions of finite presentations (preprint, University of Utah, 1991).Google Scholar
8.Gromov, M., Hyperbolic groups, in: Essays in group theory (Springer Verlag, 1987), 75–263.Google Scholar
9.Lyndon, R. and Schupp, P., Combinatorial Group Theory (Springer Verlag, Berlin, 1977).Google Scholar
10.Short, H., Groups and combings, Laboratoire de Math., Ecole Normale Sup. de Lyon (1990).Google Scholar