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Groups, Geometry, and Dynamics

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Volume 15, Issue 1, 2021, pp. 143–195
DOI: 10.4171/GGD/595

Published online: 2021-03-25

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

Motiejus Valiunas[1]

(1) University of Wroclaw, Poland

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].

Keywords: Acylindrically hyperbolic groups, equationally noetherian groups, graph products, quasi-median graphs

Valiunas Motiejus: Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products. Groups Geom. Dyn. 15 (2021), 143-195. doi: 10.4171/GGD/595