Finiteness properties of cubulated groups

TL;DR

This paper explores the construction of CAT(0) cube complexes from wallspaces, focusing on finiteness properties and group actions, especially for relatively hyperbolic groups with quasiconvex subgroups, generalizing previous results.

Contribution

It provides a generalized, self-contained framework for constructing dual cube complexes from wallspaces, extending Sageev's results to relatively hyperbolic groups with new finiteness properties.

Findings

G acts relatively cocompactly on the dual CAT(0) cube complex C.

When peripheral groups are abelian, C admits a G-cocompact CAT(0) truncation.

The framework facilitates applying wallspace methods to produce group actions on cube complexes.

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,...,H_s be relatively quasiconvex codimension-1 subgroups of a group G that is hyperbolic relative to P_1,...,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,...,P_r are abelian, we show that the dual CAT(0) cube complex C has a G-cocompact CAT(0) truncation.