Relative second bounded cohomology of free groups

Abstract

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