Cohomological lower bounds for isoperimetric functions on groups

If the finitely presented group G splits over the finitely presented sub-group C, then classes are constructed in H2(∞) (G) which reflect the splitting and which serve as lower bounds for isoperimetric functions for G. It is proved that H2(∞) (G)=0 for all word hyperbolic groups G. A converse is obtained for the combination theorem for hyperbolic groups of Bestvina–Feighn. The Mayer–Vietoris exact sequence for l∞-cohomology associated to a splitting of a group is established. Metabolic groups are introduced as finitely presented groups G such that H2(∞) (G, A)=0 for all normed abelian coefficient groups A and such groups G are shown to be characterized by possessing “thin” combings.