Finiteness properties of cubulated groups

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.