Realising VCD for untwisted automorphism groups of RAAGs

TL;DR

This paper constructs a new cube complex for untwisted automorphism groups of RAAGs, clarifying when its dimension matches the virtual cohomological dimension and demonstrating that the difference can be arbitrarily large.

Contribution

It introduces an equivariant deformation retraction of the untwisted spine to a new complex, providing conditions for dimension realization of the virtual cohomological dimension.

Findings

The new complex is contractible and admits a proper, cocompact action.

Conditions are identified under which the complex's dimension equals the vcd.

The difference between the complex's dimension and vcd can be arbitrarily large.

Abstract

The virtual cohomological dimension of~$\operatorname{Out}(F_n)$ is given precisely by the dimension of the spine of Culler--Vogtmann Outer space. However, the dimension of the spine of untwisted Outer space for a general right-angled Artin group~$A_\Gamma$ does not necessarily match the virtual cohomological dimension~$\textsc{vcd}(U(A_{\Gamma}))$ of the untwisted subgroup~$U(A_\Gamma) \leq \operatorname{Out}(A_\Gamma)$. Under certain graph-theoretic conditions, we perform an equivariant deformation retraction of this spine to produce a new contractible cube complex upon which~$U(A_\Gamma)$ acts properly and cocompactly. Furthermore, we give conditions for when the dimension of this complex realises the virtual cohomological dimension of~$U(A_\Gamma)$. We finish with two applications of our construction; in particular we show that the difference between the dimension of the untwisted spine and~$\textsc{vcd}(U(A_{\Gamma}))$ can be arbitrarily large.