Realising VCD for untwisted automorphism groups of RAAGs

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