Cube complex

A cube complex is formed by gluing Euclidean cubes along isometric faces so that intersecting cubes meet along common faces.

The link of a vertex $v$ records the local directions from $v$: its vertices are incident edge germs, and a collection spans a simplex exactly when the corresponding edge germs lie in a common cube.

A cube complex is nonpositively curved when every vertex link is a flag simplicial complex. Equivalently, its universal cover with the piecewise Euclidean metric is a CAT(0) space.

A CAT(0) cube complex is a simply connected nonpositively curved cube complex.

Two edges of a cube complex are parallel when they are opposite sides of a square, with transitive closure understood. A hyperplane is assembled from the midcubes dual to one parallelism class of edges.

The carrier $N(H)$ of a hyperplane $H$ is the union of the closed cubes that meet $H$.

A nonpositively curved cube complex is special when every hyperplane is embedded and two-sided, no hyperplane self-osculates, and no two hyperplanes interosculate. Equivalently, it admits a cubical local isometry to a Salvetti complex.

A hyperplane self-intersects when two adjacent edges of one square are dual to that same hyperplane.

A hyperplane is one-sided when its carrier is a twisted interval bundle rather than a product with an interval.

A two-sided hyperplane self-osculates when two distinct, consistently oriented edges dual to it have the same initial vertex but do not form the corner of a square.

Two hyperplanes interosculate when they cross in one square and also have dual edges meeting at a vertex without spanning a square elsewhere.

The local isometry from a connected special cube complex to a Salvetti complex induces an injection of fundamental groups. Hence its fundamental group is a subgroup of a Right-angled Artin group.

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.