We consider a finitely generated virtually abelian group acting properly and without inversions on a cube complex . We prove that stabilizes a finite-dimensional subcomplex that is isometrically embedded in the combinatorial metric. Moreover, we show that is a product of finitely many quasilines. The result represents a higher-dimensional generalization of Haglund’s axis theorem.
"A generalized axis theorem for cube complexes." Algebr. Geom. Topol. 17 (5) 2737 - 2751, 2017. https://doi.org/10.2140/agt.2017.17.2737