Groups, Geometry, and Dynamics

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Volume 11, Issue 4, 2017, pp. 1469–1495
DOI: 10.4171/GGD/435

Published online: 2017-12-07

Dimension invariants of outer automorphism groups

Dieter Degrijse[1] and Juan Souto[2]

(1) National University of Ireland, Galway, Ireland
(2) Université de Rennes 1, France

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 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$(G))-r$.

Keywords: Outer automorphism groups, geometric dimension for proper actions, virtual cohomological dimension

Degrijse Dieter, Souto Juan: Dimension invariants of outer automorphism groups. Groups Geom. Dyn. 11 (2017), 1469-1495. doi: 10.4171/GGD/435