Realising VCD for untwisted automorphism groups of RAAGs

The virtual cohomological dimension of Out(Fₙ) 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_Γ A Γ does not necessarily match the virtual cohomological dimension vcd(U(AΓ)) V C D ( U ( A Γ ) ) of the untwisted subgroup U(A_Γ ) ≤ Out(A_Γ ) U ( A Γ ) ≤ Out ( A Γ ) . 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_Γ ) U ( A Γ ) acts properly and cocompactly. Furthermore, we give conditions for when the dimension of this complex realises the virtual cohomological dimension of U(A_Γ ) U ( A Γ ) . We finish with two applications of our construction; in particular we show that the difference between the dimension of the untwisted spine and vcd(U(AΓ)) V C D ( U ( A Γ ) ) can be arbitrarily large.