= Cubical complex
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= Cube complex
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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.<Special cube complexes by Frédéric Haglund and Daniel T. Wise>[Definition 2.1, page 4]{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>.
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