Realising VCD for untwisted automorphism groups of RAAGs

Abstract

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~$A_{\Gamma }$ does not necessarily match the virtual cohomological dimension~$\textsc{vcd}(U(A_Γ))$ of the untwisted subgroup~$U(A_{\Gamma })\le \mathrm{Out}(A_{\Gamma })$. 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~$\textsc{vcd}(U(A_Γ))$ can be arbitrarily large.

v2: minor changes, incorporating referee comments. 37 pages, 13 figures. Final version, only differs stylistically from journal version. To appear in Geometriae Dedicata