Dimension invariants of outer automorphism groups

The geometric dimension for proper actions $\underline{\mathrm{gd}}(G)$ of a group $G$ is the minimal dimension of a classifying space for proper actions $\underline{E}G$. We construct for every integer $r\geq 1$, an example of a virtually torsion-free Gromov-hyperbolic group $G$ such that for every group $Γ$ which contains $G$ as a finite index normal subgroup, the virtual cohomological dimension $\mathrm{vcd}(Γ)$ of $Γ$ equals $\underline{\mathrm{gd}}(Γ)$ but such that the outer automorphism group $\mathrm{Out}(G)$ is virtually torsion-free, admits a cocompact model for $\underline E\mathrm{Out}(G)$ but nonetheless has $\mathrm{vcd}(\mathrm{Out}(G))\le\underline{\mathrm{gd}}(\mathrm{Out}(G))-r$.

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