B-Bounded cohomology and applications

TL;DR

This paper investigates conditions under which B-bounded cohomology aligns with ordinary cohomology for groups, introducing new concepts and examples, and exploring applications to various classes of groups and complexes.

Contribution

It establishes criteria for strong B-isocohomologicality, introduces a relative theory of B-bounded cohomology, and provides examples and applications for groups with specific Dehn functions.

Findings

Strong B-isocohomologicality for $FP^{inity}$ groups with bounded weighted Dehn functions

Examples where the comparison map fails to be injective or surjective

Development of a relative B-bounded cohomology theory for groups and subgroups

Abstract

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.