Outer space and finiteness properties for symmetric automorphisms of RAAGs, and generalisations

We define the symmetric (outer) automorphism group of a right-angled Artin group and construct for it a (spine of) Outer space. This `symmetric spine' is a contractible cube complex upon which the symmetric outer automorphism group acts properly and cocompactly. One artefact of our technique is a strengthening of the proof of contractibility of the untwisted spine, mimicking the original proof that Culler--Vogtmann Outer space is contractible, which may be of independent interest. We apply our results to derive finiteness properties for certain subgroups of outer automorphisms. In particular, we prove that the subgroup consisting of those outer automorphisms which permute any given finite set of conjugacy classes of a right-angled Artin group is of type \emph{VF}, and we show that the virtual cohomological dimension of the symmetric outer automorphism group is equal to both the dimension of the symmetric spine and the rank of a free abelian subgroup.