#### Volume 17, issue 5 (2017)

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A generalized axis theorem for cube complexes

### Daniel J Woodhouse

Algebraic & Geometric Topology 17 (2017) 2737–2751
##### 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.

##### Keywords
$\mathrm{CAT}(0)$ cube complexes, geometric group theory, axis
Primary: 20F65
##### Publication
Revised: 9 May 2017
Accepted: 8 April 2017
Published: 19 September 2017
##### Authors
 Daniel J Woodhouse Mathematics Department Technion – Israel Institute of Technology Haifa Israel