Automorphisms and homology of non-positively curved cube complexes

Abstract

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.

Modified the proof of Theorem 3.9 for clarity. Added Lemma 3.9. 26 pages, 1 figure