Automorphisms of graph products of groups and acylindrical hyperbolicity (Q6619637)

Let \(\Gamma=(V,E)\) be a simplicial graph and \(\mathcal{G}=\{ G_{u} \mid u \in V \}\) a collection of groups indexed by the vertices of \(\Gamma\), the graph product \(\Gamma G\) is defined as\N\[\N\Gamma G=\bigg ( \underset{u \in V}{\ast} G_{u} \bigg) \bigg / \Big \langle \Big \langle [g,h] \; \Big | \; g \in G_{u}, h \in G_{v}, (g,h) \in E \Big \rangle \Big \rangle.\N\]\NGraph products include several classical families of groups such as free products, direct products, right-angled Artin groups right-angled Coxeter groups and and several others.\N\NThis paper under review is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. The main result is that, if \(\Gamma\) is a finite graph which contains at least two vertices and is not a join and if \(\mathcal{G}\) is a collection of finitely generated irreducible groups, then either \(\Gamma G\) is infinite dihedral or \(\mathrm{Aut}(\Gamma G)\) is acylindrically hyperbolic. This theorem is new even for right-angled Artin groups and right-angled Coxeter groups. Various consequences are deduced from this statement and from the techniques used to prove it. In particular: (1) the automorphism groups of most graph products verify vastness properties such as being \(\mathsf{SQ}\)-universal; (2) many automorphism groups of graph products do not satisfy Kazhdan's property (T); (3) the isomorphism problem between graph products is, in some cases, solvable; (4) a graph product of coarse median groups, as defined by \textit{B. H. Bowditch} in [Pac. J. Math. 261, No. 1, 53--93 (2013; Zbl 1283.20048)], is coarse median itself.\N\NBelow the reviewer reproduces the content of the paper. Chapters 1 and 2: Introduction and Preliminaries. Chapter 3: A rigidity theorem (graphically irreducible groups, isomorphisms between products, proof of the rigidity theorem). Chapter 4: Generalised loxodromic elements (the small crossing graph, straight geodesics in the crossing graph, irreducible elements, WPD isometries, generalised loxodromic inner automorphisms). Chapter 5: Fixators in the outer automorphism group (tree-graded spaces, products of tree-graded spaces, asymptotic cones and Paulin's construction, products of trees of spaces, fixed-point theorems in asymptotic cones, construction of an action on a real tree). Chapter 6: Fixed point property on real trees (hyperbolic cone-offs, free subgroups avoiding parabolic subgroups, relative fixed point property). Chapter 7: Proofs of the main theorem and its corollaries. Chapter 8: A few other applications and some open questions.