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Finiteness properties of cubulated groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  10 March 2014

G. C. Hruska  and
Daniel T. Wise
G. C. Hruska
Affiliation:
Department of Mathematical Sciences, University of Wisconsin–Milwaukee, PO Box 413, Milwaukee, WI 53201, USA email chruska@uwm.edu
Daniel T. Wise
Affiliation:
Department of Mathematics & Statistics, McGill University, Montreal, QC, Canada H3A 0B9 email wise@math.mcgill.ca
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Abstract

We give a generalized and self-contained account of Haglund–Paulin’s wallspaces and Sageev’s construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let $H_1,\ldots, H_s$ be relatively quasiconvex codimension-1 subgroups of a group $G$ that is hyperbolic relative to $P_1, \ldots, P_r$. We prove that $G$ acts relatively cocompactly on the associated dual CAT(0) cube complex $C$. This generalizes Sageev’s result that $C$ is cocompact when $G$ is hyperbolic. When $P_1,\ldots, P_r$ are abelian, we show that the dual CAT(0) cube complex $C$ has a $G$-cocompact CAT(0) truncation.

Keywords

cube complexwallspacerelatively hyperbolic group

MSC classification

Secondary: 20F65: Geometric group theory 20F67: Hyperbolic groups and nonpositively curved groups
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 3 , March 2014 , pp. 453 - 506
Copyright
© The Author(s) 2014 

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