  We define an integer-valued invariant of special cube complexes called the
genus, and prove that having genus one characterizes special cube complexes
with abelian fundamental group. Using the genus, we obtain a new proof that the
fundamental group of a special cube complex is either free abelian or surjects
onto a non-cyclic free group. We also investigate automorphisms of special cube
complexes, and give a new geometric proof that the Torelli subgroup for a
right-angled Artin group is torsion-free.
