 Salvetti complex

The Salvetti complex $S_{\Gamma }$ has one vertex, one oriented loop for each vertex of $\Gamma$, and an $n$-torus cubulated by one $n$-cube for every $n$-clique of $\Gamma$. It is nonpositively curved and ${\pi }_1{S}_{\Gamma }\cong A_{\Gamma }$.