Geometriae Dedicata

, Volume 92, Issue 1, pp 41–58 | Cite as

Filling Length in Finitely Presentable Groups

  • Steve M. Gersten
  • Tim. R. Riley


We study the filling length function for a finite presentation of a group Γ, and interpret this function as an optimal bound on the length of the boundary loop as a van Kampen diagram is collapsed to the basepoint using a combinatorial notion of a null-homotopy. We prove that filling length is well behaved under change of presentation of Γ. We look at 'AD-pairs' (f,g) for a finite presentation \(\mathcal{P}\): that is, an isoperimetric function f and an isodiametric function g that can be realised simultaneously. We prove that the filling length admits a bound of the form [g+1][log (f+1)+1] whenever (f,g) is an AD-pair for \(\mathcal{P}\). Further we show that (up to multiplicative constants) if \(x^r \) is an isoperimetric function (\(r \geqslant 2\)) for a finite presentation then (\(x^r ,x^{r - 1} \)) is an AD-pair. Also we prove that for all finite presentations filling length is bounded by an exponential of an isodiametric function.

filling length finitely presented group isoperimetric function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bridson, M. R.: Asymptotic cones and polynomial isoperimetric inequalities, Topology 38(3) (1999), 543–554.Google Scholar
  2. 2.
    Bridson, M. R. and Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin, 1999.Google Scholar
  3. 3.
    Cohen, D. E.: Isoperimetric and isodiametric inequalities for group presentations, Internat. J. Algebra Comput. 1(3) (1991), 315–320.Google Scholar
  4. 4.
    Epstein, D. B. A., Cannon, J. W., Holt, S. F., Levy, S. V. F., Paterson, M. S. and Thurston, W. P.: Word Processing in Groups, Jones and Bartlett, Boston, 1992.Google Scholar
  5. 5.
    Frankel, S. and Katz, M.: The Morse landscape of a Riemannian disc, Ann. Inst. Fourier, Grenoble 43(2) (1993), 503–507.Google Scholar
  6. 6.
    Gersten, S. M.: The double exponential theorem for isoperimetric and isodiametric functions, Internat. J. Algebra Comput. 1(3) (1991), 321–327.Google Scholar
  7. 7.
    Gersten, S. M.: Isoperimetric and isodiametric functions, In: G. Niblo and M. Roller (eds), Geometric Group Theory I, London Math. Soc. Lecture Note Ser. 181, Cambridge Univ. Press, 1993.Google Scholar
  8. 8.
    Gersten, S. M.: Asynchronously automatic groups, In: Charney, Davis, and Shapiro (eds), Geometric Group Theory. Ohio State Univ. Math. Res. Inst. Publ. 3, Walter de Gruyter, Berlin, 1995, pp. 121–133.Google Scholar
  9. 9.
    Gersten, S. M. and Short, H.: Some isoperimetric inequalities for kernels of free extensions, Geom Dedicata 92 (2002), 63–73.Google Scholar
  10. 10.
    Gromov, M.: Asymptotic invariants of infinite groups, In: G. Niblo and M. Roller (eds), Geometric Group Theory II, London Math. Soc. Lecture Notes Ser. 182, Cambridge Univ. Press, 1993.Google Scholar
  11. 11.
    Lyndon, R. C. and Schupp, P. E.: Combinatorial Group Theory, Ergeb. Math. Grenzgeb. 89, Springer-Verlag, New York, 1977.Google Scholar
  12. 12.
    Papasoglu, P.: On the asymptotic invariants of groups satisfying a quadratic isoperimetric inequality, J. Differential Geom. 44 (1996), 789–806.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Steve M. Gersten
    • 1
  • Tim. R. Riley
    • 2
  1. 1.Mathematics DepartmentUniversity of UtahSalt Lake CityU.S.A.
  2. 2.Mathematical InstituteOxfordU.K.

Personalised recommendations