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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.

For a finite graph $\Gamma$, its Right-angled Artin group isIts Salvetti complex $S_{\Gamma }$ is the one-vertex cube complex with an oriented loop labelled $v$ for each vertex of $\Gamma$, a square torus for every edge, and more generally one cubulated $k$-torus for every $k$-clique, attached compatibly along coordinate subtori. Thus ${\pi }_1(S_{\Gamma })\cong A_{\Gamma }$. The vertex link of a Salvetti complex is flag, so $S_{\Gamma }$ is a nonpositively curved cube complex.

If $\Gamma ={\Gamma }_1\bigsqcup {\Gamma }_2$ is a nontrivial disjoint union of graphs, no defining relation mixes the two vertex sets, and henceOn the topological side, every clique lies in one component, soa one-point union of Salvetti complexes.

If $\Gamma ={\Gamma }_1\ast {\Gamma }_2$ is a nontrivial join of graphs, every generator from the first part commutes with every generator from the second. ThereforeEvery clique of the join is the union of a clique in each factor, which gives the cubical identity

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.