Acylindrical actions of groups on trees

Abstract

We develop the theory of groups that act acylindrically and non-elementarily on simplicial trees, which we call acylindrically arboreal groups. We prove a complete classification of when graph products of groups and the fundamental groups of certain hyperbolic 3-manifolds are acylindrically arboreal, and use these classifications to provide examples of acylindrically hyperbolic groups that have non-trivial actions on trees but have no non-elementary acylindrical actions on trees. Further, we give a necessary and sufficient condition for a visual splitting of an Artin group satisfying the conclusions of two well known conjectures to correspond to an acylindrical action on a tree, and demonstrate how this can be used to provide a large class of novel examples of Artin groups that satisfy the Tits alternative. We also consider dominations of acylindrical actions on trees. We find a condition on an acylindrical action of a finitely presented group on a simplicial tree which guarantees that this action will be dominated by an acylindrical action with finitely generated edge stabilisers, and find the first example of an action of a finitely presented group where there is no such dominating action. As a consequence, we show that any finitely presented group acting acylindrically on a tree with (possibly infinitely generated) free groups is virtually compact special, and that every finitely generated subgroup of a one-relator group with an acylindrical Magnus hierarchy is virtually compact special.