Relative second bounded cohomology of free groups

TL;DR

This paper computes the relative second bounded cohomology of free groups with respect to subgroups, establishing a criterion for non-triviality based on subgroup index and embedding properties.

Contribution

It provides a precise characterization of when the relative second bounded cohomology is non-trivial for free groups and describes its structure via isometric embeddings.

Findings

H_b^2(\Gamma,H;\mathbb{R}) is non-trivial iff H has infinite index in \\Gamma

The cohomology space contains an isometric embedding of a direct sum of defect spaces

The result links subgroup index to the structure of bounded cohomology in free groups.

Abstract

This paper is devoted to the computation of the space $H_b^2(\Gamma,H;\mathbb{R})$, where $\Gamma$ is a free group of finite rank $n\geq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in $\Gamma$ if and only if $H_b^2(\Gamma,H;\mathbb{R})$ is not trivial, and furthermore, if and only if there is an isometric embedding $\oplus_\infty^n\mathcal{D}(\mathbb{Z})\hookrightarrow H_b^2(\Gamma,H;\mathbb{R})$, where $\mathcal{D}(\mathbb{Z})$ is the space of bounded alternating functions on $\mathbb{Z}$ equipped with the defect norm.