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
Download PDF

This article is published open access under our Subscribe to Open model.

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 and define the contact graph for these hyperplanes. We show that is always quasi-isometric to a tree, generalising a result of Hagen [18], 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 [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 , 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 [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