The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Groups, Geometry, and Dynamics


Full-Text PDF (540 KB) | Metadata | Table of Contents | GGD summary
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