Wallspace

A wallspace $(S,\mathcal{W})$ is a set $S$ with walls $W=\{W^{+},W^{-}\}$, each a partition into two halfspaces, such that only finitely many walls separate any two points of $S$.

A wall is an unordered pair of complementary halfspaces $\{W^{+},W^{-}\}$.

A halfspace is one of the two complementary parts determined by a wall.

Two walls cross when all four intersections of one halfspace from each wall are nonempty. A collection of pairwise crossing walls is also called transverse.

A transverse collection of walls is a collection in which every two distinct walls cross.

The wall metric is

The vertices of the dual cube complex are consistent choices of one halfspace of every wall that differ from a principal choice on only finitely many walls. Two vertices are joined when their choices differ on one wall, and higher cubes fill the resulting hypercubes. The result is a CAT(0) cube complex.

A point $x\in S$ determines a principal vertex by choosing, for every wall, the halfspace containing $x$. The combinatorial distance between the principal vertices of $x$ and $y$ is $d_{\mathcal{W}}(x,y)$.

A cubulation of a group is a metrically proper action by cubical automorphisms on a CAT(0) cube complex. It is cocompact when the action is also a cocompact group action.

An isometric group action on a metric space is metrically proper when, for some and hence every point $x$, the set $\{g:d(x,gx)\le R\}$ is finite for every finite $R$.

If $G$ acts on $(S,\mathcal{W})$ and $x\in S$, then its action on the dual cube complex is metrically proper wheneveras $g$ leaves every finite subset of $G$.

The action of $G$ on the dual cube complex is cocompact exactly when there are finitely many $G$-orbits of finite transverse wall collections. Equivalently, there is a uniform bound on their sizes and, for every size, only finitely many orbits.