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Geometriae Dedicata

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

Filling Length in Finitely Presentable Groups

  • Steve M. Gersten
  • Tim. R. Riley
Article

Abstract

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 

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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.

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