A generalized axis theorem for cube complexes

Abstract

We consider a finitely generated virtually abelian group $G$ acting properly and without inversions on a $CAT\left(0\right)$ cube complex $X$. We prove that $G$ stabilizes a finite-dimensional $CAT\left(0\right)$ subcomplex $Y\subseteq X$ that is isometrically embedded in the combinatorial metric. Moreover, we show that $Y$ is a product of finitely many quasilines. The result represents a higher-dimensional generalization of Haglund’s axis theorem.