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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.

Daniel T. Wise studies geometric group theory through cubical geometry. With Frédéric Haglund, he introduced and studied special cube complexes, whose hyperplanes satisfy restrictions preventing certain self-interactions.^[ref]

A compact nonpositively curved cube complex satisfying their criterion admits a local isometry into the standard complex of a Right-angled Artin group. Consequently its fundamental group embeds in a linear group. The contribution connects visible geometric conditions on hyperplanes with algebraic properties of the fundamental group.^[ref]

Frédéric Haglund works on geometric group theory and cubical geometry. His joint work with Daniel T. Wise introduced special cube complexes and related them to Right-angled Artin groups.^[ref]

The key conditions control how hyperplanes embed, whether they are two-sided and how they meet. A local isometry to a standard cubical model then yields an embedding of the fundamental group. This is a concrete mechanism for obtaining algebraic information from the internal geometry of a complex.^[ref]