Bounded cohomology and virtually free hyperbolically embedded subgroups

TL;DR

The paper demonstrates that for acylindrically hyperbolic groups, the second bounded cohomology can be embedded into the inverse limit of the second bounded cohomologies of its virtually free hyperbolically embedded subgroups, revealing new structural insights.

Contribution

It introduces a probabilistic method to embed the second bounded cohomology of acylindrically hyperbolic groups into inverse limits of their virtually free subgroups' cohomologies, a novel result.

Findings

Embedding of second bounded cohomology into inverse limits

Applicability to various hyperbolic and relatively hyperbolic groups

Failure of similar results for third bounded cohomology

Abstract

Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${\rm Out}(F_n)$, \dots) embeds via the natural restriction maps into the inverse limit of the second bounded cohomologies of its virtually free subgroups, and in fact even into the inverse limit of the second bounded cohomologies of its hyperbolically embedded virtually free subgroups. This result is new and non-trivial even in the case where $G$ is a (non-free) hyperbolic group. The corresponding statement fails in general for the third bounded cohomology, even for surface groups.