
Filling length in finitely presented groupsSteve M. Gersten and Tim R. RileyGeometriae 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 nullhomotopy. 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 ADpair for P. Further we show that (up to multiplicative constants) if x^{r} is an isoperimetric function (r \geq 2) for a finite presentation then (x^{r},x^{r1}) is an ADpair. Also we prove that for all finite presentations filling length is bounded by an exponential of an isodiametric function.
