Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T15:53:09.612Z Has data issue: false hasContentIssue false

Annular Dehn functions of groups

Published online by Cambridge University Press:  17 April 2009

Stephen G. Brick
Affiliation:
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, United States of America e-mail: [email protected]
Jon M. Corson
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa AL 35487-0350, United States of America 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.

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.

As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Alonso, J.M., ‘Inegalités isopérimétriques et quasi-isométries’, C.R. Acad. Sci. Paris 311 (1990), 761764.Google Scholar
[2]Brick, S.G., ‘On Dehn functions and products of groups’, Trans. Amer. Math. Soc. 335 (1993), 369384.CrossRefGoogle Scholar
[3]Collins, D. and Miller, C.F. III, ‘The conjugacy problem and subgroups of finite index’, Proc. London Math. Soc. 34 (1977), 535556.CrossRefGoogle Scholar
[4]Juhàsz, A., ‘Solution of the conjugacy problem in one-relator groups’, in Proceedings of the Workshop on Algorithmic Problems, Math. Sci. Res. Inst. Publ. 23 (Springer-Verlag, Berlin, Heidelberg, New York, 1991).Google Scholar
[5]Lyndon, R.C. and Schupp, P.E., Combinatorial group theory (Springer-Verlag, Berlin, Heidelberg, New York, 1977).Google Scholar
[6]Neumann, W.D., ‘Asynchronous combings of groups’, Intnat. J. Algebra Comput. 2 (1992), 179185.CrossRefGoogle Scholar
[7]Pride, S.J., ‘Star-complexes, and the dependence problems for hyperbolic complexes’, Glasgow Math. J. 30 (1988), 155170.CrossRefGoogle Scholar