Cubical complex

A cube complex, in the cubical CW complex convention used here, is built from standard hypercubes $[0,1{]}^n$ by identifying their faces of a cube using isometries. The restriction of each characteristic map to a face of a cube agrees with the characteristic map of a lower-dimensional cell, after an isometry of that face. The cubes carry their Euclidean metrics, giving an induced length distance on each connected component.^[ref]

Face identifications may identify different boundary points of the same cube. For example, one interval with its endpoints identified is a circle, and one square with opposite sides identified is a torus. The latter has one vertex, two edges and one square; it is the Salvetti complex for two commuting generators.

A stricter polyhedral convention requires every closed cube to embed and every pair of cubes to meet in one common face. That convention needs a finer cell decomposition of these examples. The cubical-CW convention allows the one-vertex quotient models used in geometric group theory. Being a nonpositively curved cube complex is an additional condition, detected by the link of a vertex in a cube complex.

A $k$-dimensional face of the standard $n$-cube $[0,1{]}^n$ is obtained by choosing $k$ coordinates to vary freely and fixing each of the other $n-k$ coordinates to either zero or one. It is therefore isometric to $[0,1{]}^k$.

There arefaces of dimension $k$: choose the free coordinates and then their complementary endpoint values. For the three-dimensional cube, this gives eight vertices, twelve edges and six square faces. The full cube is the unique $n$-dimensional face. Some conventions also adjoin an empty face of dimension $-1$.

In a cubical complex, characteristic maps attach these faces to lower-dimensional cubes by isometries. Distinct abstract faces can be identified in the quotient complex, so a cell's face count in its model cube need not equal the number of distinct lower-dimensional cells in the whole complex.

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 $S$.

Such wall partitions encode separation data used to construct a dual cube complex.

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.

For a wallspace $(S,\mathcal{W})$, defineThe wallspace axiom makes this finite. It is always a pseudometric: it is symmetric, vanishes on the diagonal and satisfies the triangle inequality. For the last claim, every wall separating $x$ and $z$ must separate $x$ from $y$ or $y$ from $z$, so counting walls gives the inequality.^[ref]

It is a metric exactly when the walls separate every pair of distinct points. For example, on $S=\{a,b,c\}$ with only the wall $\{\{a,b\},\{c\}\}$, the distinct points $a,b$ have distance zero. Identifying pairs at distance zero gives a genuine metric space.

The principal vertex of a dual cube complex sends indistinguishable points to the same vertex. The wall-count distance equals the combinatorial distance between their principal vertices, making this pseudometric useful for the wall-metric properness criterion.

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.