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.
Categories: math.GR math.GT