Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products

In this paper we study group actions on quasi-median graphs, or 'CAT(0) prism complexes', generalising the notion of CAT(0) cube complexes. We consider hyperplanes in a graph $X$ and define contact $\mathcal{C}X$ for these hyperplanes. show that is always quasi-isometric to tree, result Hagen, under certain conditions action $G \curvearrowright X$ induces an acylindrical \mathcal{C}X$, giving analogue Behrstock, Hagen Sisto. As application, exhibit product quasi-tree, results Kim Koberda right-angled Artin groups. many products $G$, 'largest' $G$ hyperbolic metric space. use equationally noetherian groups over finite graphs girth $\geq 6$ are noetherian, Sela.