When is the graph product of hyperbolic groups hyperbolic? (Q1914049)

Given a finite simplicial graph \({\mathcal G}\) and a group \(G_v\) assigned to each vertex \(v\), the graph product \(G{\mathcal G}\) is the free product of the vertex groups with added relations that imply elements of adjacent vertex groups commute. Given a vertex \(v\) of \(\mathcal G\), the link graph of \(v\), denoted by \({\mathcal L}_v\), is the subgraph of \(\mathcal G\) generated by the vertices of \(\mathcal G\) adjacent to \(v\). The author proves the following Theorem: Let \({\mathcal G}\) be a simplicial graph with word hyperbolic groups assigned to its vertices. Let \({\mathcal F}_{\mathcal G}\) be the full subgraph of \(\mathcal G\) generated by the vertices associated with finite groups. Then, the graph product \(G{\mathcal G}\) is word hyperbolic if and only if the following three conditions hold: (i) the full subgraph generated by the vertices in \({\mathcal G}-{\mathcal F}_{\mathcal G}\) is a null-graph; (ii) for \(v\in{\mathcal G}-{\mathcal F}_{\mathcal G}\), \({\mathcal L}_v\) is a complete graph; (iii) every circuit in \({\mathcal F}_{\mathcal G}\) of length four contains a chord. The main point in the proof involves the construction of a \(\text{CAT}(-1)\) cubical complex admitting a discrete, cocompact action of \(G{\mathcal F}_{\mathcal G}\).