Filling length in finitely presented groups

Steve M. Gersten and Tim R. Riley

Geometriae Dedicata, 92(1), pages 41–58, July 2002.

We study the filling length function for a finite presentation of a group \Gamma, 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 \Gamma.

We look at "AD–pairs" (f,g) for a finite presentation 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 P. Further we show that (up to multiplicative constants) if xr is an isoperimetric function (r \geq 2) for a finite presentation then (xr,xr-1) is an AD-pair. Also we prove that for all finite presentations filling length is bounded by an exponential of an isodiametric function.

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