Automorphisms of Graph Products of Groups and Acylindrical Hyperbolicity

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite graph which contains at least two vertices and is not a join and if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a collection of finitely generated irreducible groups, then either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma script upper G"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \mathcal {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is infinite dihedral or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper A normal u normal t left-parenthesis normal upper Gamma script upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">A</mml:mi> <mml:mi mathvariant="normal">u</mml:mi> <mml:mi mathvariant="normal">t</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {Aut}(\Gamma \mathcal {G})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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. For instance, we show that the automorphism groups of most graph products verify vastness properties such as being SQ-universal; we show that many automorphism groups of graph products do not satisfy Kazhdan’s property (T); we solve the isomorphism problem between graph products in some cases; and we show that a graph product of coarse median groups, as defined by Bowditch, is coarse median itself.