Bounded acyclicity and relative simplicial volume

Abstract

We provide new vanishing and glueing results for relative simplicial volume, following up on two current themes in bounded cohomology: The passage from amenable groups to boundedly acyclic groups and the use of equivariant topology.More precisely, we consider equivariant nerve pairs and relative classifying spaces for families of subgroups. Typically, we apply this to uniformly boundedly acyclic families of subgroups. Our methods also lead to vanishing results for ℓ 2 -Betti numbers of aspherical CW-pairs with small relative amenable category and to a relative version of a result by Dranishnikov and Rudyak concerning mapping degrees and the inheritance of freeness of fundamental groups.

Bounded Cohomology and Binate Groups

“…Here S is the operator norm of S with respect to the 1 -norms. 6 F. Fournier-Facio C. Löh and M. Moraschini [6] The uniform boundary condition will be more systematically reviewed in a forthcoming paper [38].…”

“…The effect of boundedly acyclic stabilizers is studied more systematically in a forthcoming article on boundedly acyclic covers and relative simplicial volume [38].…”

The spectrum of simplicial volume with fixed fundamental group

“…Good bounds for such amenable multiplicities are, e.g., known for right-angled Artin groups [16]. More generally, one can also consider multiplicities of (uniformly) boundedly acyclic open covers [15,17].…”

“…The statement of Lemma 4.2 can be generalised to uniformly boundedly acyclic edge groups by using the corresponding result on bounded cohomology of such graphs of groups [17,Theorem 8.11].…”