Right-angled Artin group ($A_{\Gamma }$)

For a graph $\Gamma$, the right-angled Artin group is

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 ${\pi }_1{S}_{\Gamma }\cong A_{\Gamma }$.

The unique vertex of $S_{\Gamma }$ has two link vertices $v^{+},v^{-}$ for each $v\in V(\Gamma )$. Signed vertices $v_1^{{ϵ}_1},\dots ,v_k^{{ϵ}_k}$ span a simplex exactly when the distinct underlying vertices $v_1,\dots ,v_k$ form a clique of $\Gamma$.

The hyperplane of $S_{\Gamma }$ dual to the loop labelled $v$ is naturally the Salvetti complex of the induced subgraph on the neighbours of $v$.

Every right-angled Artin group is a residually finite group. One route is its faithful integral linear representation, followed by reduction modulo suitable primes.