Bounded cohomology of classifying spaces for families of subgroups

TL;DR

This paper develops a bounded cohomology theory for classifying spaces relative to subgroup families, providing new characterizations of properties like amenability and hyperbolicity in a bounded cohomological framework.

Contribution

It introduces a bounded Bredon cohomology for groups relative to subgroup families, generalizing existing bounded cohomology theories and offering new cohomological characterizations of group properties.

Findings

Cohomological characterizations of relative amenability.

Cohomological characterizations of relative hyperbolicity.

Generalization of bounded cohomology to classifying spaces for families.

Abstract

We introduce a bounded version of Bredon cohomology for groups relative to a family of subgroups. Our theory generalizes bounded cohomology and differs from Mineyev--Yaman's relative bounded cohomology for pairs. We obtain cohomological characterizations of relative amenability and relative hyperbolicity, analogous to the results of Johnson and Mineyev for bounded cohomology.