Filling length in finitely presentable groups. Dedicated to John Stallings on the occasion of his 65th birthday

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 vanKampen 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 cal 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 cal P. Further we show that (up to multiplicative constants) if x(r) is an isoperimetric function (r greater than or equal to 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.