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

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 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 [Hag14], and that under certain conditions a group action G X induces an acylindrical action G CX, giving a quasi-median analogue of a result of Behrstock, Hagen and Sisto [BHS17].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 [KK13, KK14]. 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 ≥ 5 are equationally noetherian, generalising a result of Sela [Sel10]. Contents 1. Introduction 1 2. Preliminaries 4 3. Geometry of the contact graph 8 4. Acylindricity 11 5. Application to graph products 17 6. Equational noetherianity of graph products 20 References 28

Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups

“…Osin in [Osi16] shows that the existence of an acylindrical action of a group G on some hyperbolic space is equivalent to the existence of a hyperbolically embedded subgroup, or of a weakly properly discontinuous element for the action of G on some possibly different hyperbolic space, clarifying the existence of an extremely natural class of acylindrically hyperbolic groups. For such a group, Abbott-Balasubramanya-Osin explore the poset of acylindrical actions on hyperbolic spaces in [ABO19]; this poset is further explored in [Abb16, ABB + 17, Bal19,Val20].…”

On the geometry of van Kampen diagrams of graph products of groups

“…As a final remark, we have to mention that a simple proof of the normal form of graph products can also be found in [HW99, Section 4]. Also, the notion of dual curves we use in this article have been introduced recently and independently in [Val18]. In particular, Proposition 1.1 can be thought of as a generalisation of [Val18,Lemma 2.3].…”

“…Also, the notion of dual curves we use in this article have been introduced recently and independently in [Val18]. In particular, Proposition 1.1 can be thought of as a generalisation of [Val18,Lemma 2.3].…”