JournalsggdVol. 15, No. 1pp. 143–195

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

  • Motiejus Valiunas

    University of Wroclaw, Poland
Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products cover
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Abstract

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 XX and define the contact graph CX\mathcal{C}X for these hyperplanes. We show that CX\mathcal{C}X is always quasi-isometric to a tree, generalising a result of Hagen [18], and that under certain conditions a group action GXG \curvearrowright X induces an acylindrical action GCXG \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 GG, the action we exhibit is the "largest" acylindrical action of GG on a hyperbolic metric space. We use this to show that the graph products of equationally noetherian groups over finite graphs of girth 6\geq 6 are equationally noetherian, generalising a result of Sela [27].

Cite this article

Motiejus Valiunas, Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products. Groups Geom. Dyn. 15 (2021), no. 1, pp. 143–195

DOI 10.4171/GGD/595