Dimension invariants of outer automorphism groups

The geometric dimension for proper actions $\underset{‾}{\mathrm{g}\mathrm{d}}(G)$ of a group $G$ is the minimal dimension of a classifying space for proper actions $\underset{‾}{E}G$. We construct for every integer $r\ge 1$, an example of a virtually torsion-free Gromov-hyperbolic group $G$ such that for every group $\mathrm{\Gamma }$ which contains $G$ as a finite index normal subgroup, the virtual cohomological dimension $\mathrm{v}\mathrm{c}\mathrm{d}(\mathrm{\Gamma })$ of $\mathrm{\Gamma }$ equals $\underset{‾}{\mathrm{g}\mathrm{d}}(\mathrm{\Gamma })$ but such that the outer automorphism group $\mathrm{O}\mathrm{u}\mathrm{t}(G)$ is virtually torsion-free, admits a cocompact model for $\underset{‾}{E}\mathrm{O}\mathrm{u}\mathrm{t}(G)$ but nonetheless has $\mathrm{v}\mathrm{c}\mathrm{d}(\mathrm{O}\mathrm{u}\mathrm{t}(G))\le \underset{‾}{\mathrm{g}\mathrm{d}}(\mathrm{O}\mathrm{u}\mathrm{t}(G))-r$.