A Generalized Axis Theorem for Cube Complexes

TL;DR

This paper generalizes Haglund's axis theorem to higher dimensions, proving that finitely generated virtually abelian groups acting on CAT(0) cube complexes stabilize a product of quasilines within a finite-dimensional subcomplex.

Contribution

It introduces a higher-dimensional axis theorem for CAT(0) cube complexes, extending Haglund's result to more complex group actions.

Findings

Virtually abelian groups stabilize finite-dimensional subcomplexes.

The stabilized subcomplexes are products of quasilines.

The result generalizes Haglund's axis theorem to higher dimensions.

Abstract

We consider a finitely generated virtually abelian group $G$ acting properly and without inversions on a CAT(0) cube complex $X$. We prove that $G$ stabilizes a finite dimensional CAT(0) 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.