Dimension invariants of outer automorphism groups

Abstract

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 $r\ge 1$, an example of a virtually torsion-free Gromov-hyperbolic group $G$ such that for every group $\Gamma$ which contains $G$ as a finite index normal subgroup, the virtual cohomological dimension vcd$(\Gamma )$ of $\Gamma$ equals $\underline{\mathrm{gd}}(\Gamma )$ but such that the outer automorphism group Out$(G)$ is virtually torsion-free, admits a cocompact model for $\underline{E}$ Out$(G)$ but nonetheless has vcd(Out$(G))\le \underline{\mathrm{gd}}$(Out$(G))-r$.