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A generalized axis theorem for cube complexes

Daniel J Woodhouse

Algebraic & Geometric Topology 17 (2017) 2737–2751

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 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.

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$\mathrm{CAT}(0)$ cube complexes, geometric group theory, axis
Mathematical Subject Classification 2010
Primary: 20F65
Received: 3 February 2016
Revised: 9 May 2017
Accepted: 8 April 2017
Published: 19 September 2017
Daniel J Woodhouse
Mathematics Department
Technion – Israel Institute of Technology