A cube complex is just what it sounds like: a collection of cubes, except that the word “cube” refers not just to the usual three-dimensional cube but to the shape in any dimension consisting of all the points whose coordinates lie between, say, -1 and +1. For example, a square is considered a two-dimensional cube, and a line segment is a one-dimensional cube. The cubes in a cube complex are connected to each other along corners, edges, faces and higher-dimensional sides.
Cube complexes are very different creatures from three-manifolds — they aren’t even manifolds, for starters, since the junctions between two cubes of different dimensions don’t resemble ordinary space of any dimension. Yet cube complexes are a simplified setting in which to study one key aspect of a surface sitting inside a three-manifold: the fact that such a surface, at least locally, divides its surroundings into two sides.
If your goal is to study objects that divide a shape into two sides, cubes are a natural place to start, since of all possible shapes, they have some of the simplest such objects: the “hyperplanes” that cut across the middle of the cube. A square has two hyperplanes — the vertical and horizontal lines that each chop the square in half — and a cube has three hyperplanes (see Figure 9). An n-dimensional cube has n hyperplanes, which all intersect at the center point of the cube.
“The hyperplanes are like surfaces in a three-manifold, but you see them immediately,” Wise said. “Finding surfaces is hard, but hyperplanes are available to you right to begin with.”
If we start with a hyperplane inside a cube in a cube complex, there is exactly one way to extend the hyperplane to hyperplanes in the adjacent cubes; after that, there’s exactly one way to extend those hyperplanes to their adjacent cubes; and so on. Thus, given a starting hyperplane in a cube complex, there’s a unique way to extend it to a hyperplane in the full cube complex (see Figure 10).
This quality provides a stark contrast to three-manifolds, in which a small piece of surface can be extended in any number of ways to a full surface. Cube complexes and their hyperplanes are “nice and crystalline and rigid,”Agol said, with none of the “flabbiness” of a three-manifold and its surfaces.
As we extend a hyperplane through a cube complex, it may come back to the cube where it started and pass through it along a hyperplane perpendicular to the original one (see Figure 11). In other words, the extended hyperplane might not be embedded. Just as with surfaces inside three-manifolds, we can ask whether the cube complex has a finite cover in which these self-inteurse of the last decade, Wise has developed an arsenal of techniques for figuring out which cube complexes are special. In 2009, Wise circulated a 200-page “masterwork,” as Dunfield put it, in which he detailed a host of findings about special cube complexes, such as his “combination theorems,” which show how to piece together special cube complexes to get new ones that are guaranteed to still be virtually special. In this paper, Wise formulated a conjecture that stated, very roughly, that any cube complex whose geometry curves around in a similar way to hyperbolic geometry is “virtually” special — that is, it has a special finite cover. This statement came to be known as Wise’s conjecture.
Wise was convinced that when a given shape is akin to a cube complex in a particular way — that is, when it can be “cubulated” — the structure of the cube complex is the key to unlocking many attributes of the original shape.
“The cube complex was a secret that people didn’t even know to ask about,” he said. “It’s a fundamental, hidden intrinsic structure.”