Cube complexes and abelian subgroups of automorphism groups of RAAGs

We construct free abelian subgroups of the group $U(A_\Gamma)$ of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group $U(A_\Gamma)$ was previously studied by Charney, Stambaugh and the second author, who constructed a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of the principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.
Comment: Improvements in exposition, Lemma 4.27 added to clarify proof of Theorem 4.28, remark 3.8 about the definition of "compatibility" added