${\mathcal{B}$-BOUNDED COHOMOLOGY AND APPLICATIONS

Abstract

A discrete group with word-length (G, L) is [Formula: see text]-isocohomological for a bounding classes [Formula: see text] if the comparison map from [Formula: see text]-bounded cohomology to ordinary cohomology (with coefficients in ℂ) is an isomorphism; it is strongly [Formula: see text]-isocohomological if the same is true with arbitrary coefficients. In this paper we establish some basic conditions guaranteeing strong [Formula: see text]-isocohomologicality. In particular, we show strong [Formula: see text]-isocohomologicality for an FP∞ group G if all of the weighted G-sensitive Dehn functions are [Formula: see text]-bounded. Such groups include all [Formula: see text]-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 [Formula: see text]-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 relativeFP∞ property for groups with respect to a collection of subgroups. Applications for computing [Formula: see text]-bounded cohomology of groups are given in the context of relatively hyperbolic groups and developable complexes of groups.