Geometriae Dedicata

, Volume 92, Issue 1, pp 41–58

# 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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