Recently, the geometry of CAT(0) cube complexes featured prominently in Agol’s resolution of two longstanding conjectures of Thurston in low-dimensional topology: the virtually Haken and virtually fibered conjecture for hyperbolic 3-manifolds. A key step of the proof was to show that every hyperbolic 3-manifold group is virtually special, i.e. virtually the fundamental group of a special nonpositively curved (NPC) cube complex. In this thesis, we study algebraic properties of special groups as they relate to the geometry of special cube complexes.

In the first part of the thesis, we introduce a positive integer-valued invariant of special cube complexes called the genus, and show that having genus one is equivalent to having free abelian fundamental group. As a corollary, we obtain a new proof of the fact that every special group is either abelian or surjects onto a non-abelian free group. In the second part of the thesis, we turn our attention to automorphisms of NPC cube complexes. We give a criterion on a special cube complex which implies that any automorphism acts non-trivially on first homology, and show that a non- trivial action on homology can always be achieved by passing to covers. We then apply the criterion to provide a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.