Homological Dehn Functions and the Word Problem

S. M. Gersten Abstract. Homological Dehn functions over R and over Z are introduced to measure minimal fillings of integral 1-cycles by (real or integral) 2-chains in the Cayley 2complex of a finitely presented group. If the group G is the fundamental group of a finite graph of finitely presented vertex-groups Hv and finitely generated edge-groups, then there is a formula for an isoperimetric function (for genus 0 fillings) for G in terms of the real homological Dehn function for G and the (genus 0) Dehn functions for the Hv . Hierarchies of groups are introduced in which an isoperimetic function is determined by a formula in terms of the real homological Dehn function. In such a hierarchy the word problem is one of homological algebra. All 1-relator groups are in such a hierarchy. Applications are given to the generalized word problem (a.k.a. membership or Magnus problem) and to a homolgical determination of distortion.