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.
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