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 $X$ and define the contact graph $\mathcal{C}X$ for these hyperplanes. We show that $\mathcal{C}X$ is always quasi-isometric to a tree, generalising a result of Hagen [18], and that under certain conditions a group action $G \curvearrowright X$ induces an acylindrical action $G \curvearrowright \mathcal{C}X$, giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto [5].

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 $G$, the action we exhibit is the "largest" acylindrical action of $G$ on a hyperbolic metric space. We use this to show that the graph products of equationally noetherian groups over finite graphs of girth $\geq 6$ are equationally noetherian, generalising a result of Sela [27].

中文翻译：

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作用于图产品中拟中图和方程组的圆柱双曲性

在本文中，我们研究了准中图或“ CAT（0）棱镜复合体”上的群作用，概括了CAT（0）立方复合体的概念。我们在准中值图$ X $中考虑超平面，并为这些超平面定义接触图$ \ mathcal {C} X $。我们证明$ \ mathcal {C} X $总是对树是准等距的，推广了Hagen [18]的结果，并且在某些条件下，群体作用$ G \ curvearrowright X $引起了圆柱作用$ G \ curvearrowright \ mathcal {C} X $，给出贝斯托克，哈根和西斯托[5]的结果的准中值类似物。

作为一种应用，我们展示了图积在准树上的柱面作用，归纳了Kim和Koberda对直角Artin组的结果[20，21]。我们表明，对于许多图形产品$ G $，我们展示的作用是双曲线度量空间上$ G $的“最大”圆柱作用。我们用它来表明，周长为$ \ geq 6 $的有限图上的方程式noetherian基团的图形乘积为方程式noetherian，推广了Sela [27]的结果。