Dimension invariants of outer automorphism groups

TL;DR

This paper constructs examples of hyperbolic groups where the geometric dimension for proper actions and virtual cohomological dimension coincide for the group and its finite index supergroups, but the outer automorphism groups exhibit a gap between these dimensions.

Contribution

It provides the first examples demonstrating that the virtual cohomological dimension of outer automorphism groups can be strictly less than their geometric dimension, even when the base group has matching dimensions.

Findings

Constructed hyperbolic groups with specific dimension properties.

Showed outer automorphism groups can have lower virtual cohomological dimension.

Demonstrated the existence of groups with a gap between ext{vcd} and ext{gd} for outer automorphisms.

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 $r\geq 1$, 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 $\mathrm{vcd}(\Gamma)$ of $\Gamma $ equals $\underline{\mathrm{gd}}(\Gamma)$ 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 $\mathrm{vcd}(\mathrm{Out}(G))\le\underline{\mathrm{gd}}(\mathrm{Out}(G))-r$.