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September 23, 2025Open Access

Groups acting acylindrically on trees

Key Points

The classification reveals conditions for graph products to be acylindrically arboreal, influencing further research.
It identifies specific acylindrically hyperbolic groups with tree actions but lacking non-elementary acylindrical actions.
The study introduces acylindrically arboreal groups, expanding the understanding of group actions on trees.
It emphasizes the uniqueness of actions on trees for certain hyperbolic 3-manifold fundamental groups.

Abstract

Abstract We develop a notion of groups that act acylindrically and non-elementarily on simplicial trees, which we call acylindrically arboreal groups. We then 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 actions on trees but have no non-elementary acylindrical actions on trees.

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