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(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.

14 pages Corrected proof of Corollary 1.4. Various other corrections made following referee report and comments made by thesis examiner. Appendix added giving a proof of a theorem by Gerasimov

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