Relative cohomological dimension of a relatively hyperbolic pair

We show that the relative cohomological dimension $\text{cd}(G,H)$ of a relatively hyperbolic pair $(G,H)$ is always finite when $G$ is torsion-free. We also show that this dimension is preserved under quasi-isometries, provided that $G$ is torsion-free and the peripheral subgroup $H$ is unconstricted and of type $F_{\mathrm{\infty }}$. As a corollary of our methods, we compute $\text{cd}(G,H)$ in a range of cases.