B-Bounded cohomology and applications

A discrete group with word-length (G,L) is B-isocohomological for a bounding classes B if the comparison map from B-bounded cohomology to ordinary cohomology (with complex coefficients) is an isomorphism; it is strongly B-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong B-isocohomologicality. In particular, we show strong B-isocohomologicality for an $FP^{\infty}$ group G if all of the weighted G-sensitive Dehn functions are B-bounded. Such groups include all B-asynchronously combable groups; moreover, the class of such groups is closed under constructions arising from groups acting on an acyclic complex. We also provide examples where the comparison map fails to be injective, as well as surjective, and give an example of a solvable group with quadratic first Dehn function, but exponential second Dehn function. Finally, a relative theory of B-bounded cohomology of groups with respect to subgroups is introduced. Relative isocohomologicality is determined in terms of a new notion of relative Dehn functions and a relative $FP^\infty$ property for groups with respect to a collection of subgroups. Applications for computing B-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.