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

摘要

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 CX for these hyperplanes. We show that CX is always quasi-isometric to a tree, generalising a result of Hagen [18], and that under certain conditions a group action G curved right arrow X induces an acylindrical action G curved right arrow CX, giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto [5].