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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 quasi-median graph and define the contact graph for these hyperplanes. We show that is always quasi-isometric to a tree, generalising a result of Hagen , and that under certain conditions a group action induces an acylindrical action , giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto .
As an application, we exhibit an acylindrical action of a graph product on a quasi-tree, generalising results of Kim and Koberda for right-angled Artin groups [20, 21]. We show that for many graph products , the action we exhibit is the "largest" acylindrical action of on a hyperbolic metric space. We use this to show that the graph products of equationally noetherian groups over finite graphs of girth are equationally noetherian, generalising a result of Sela .
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Motiejus Valiunas, Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products. Groups Geom. Dyn. 15 (2021), no. 1, pp. 143–195DOI 10.4171/GGD/595