## Acylindrical group actions on quasi-trees.(English)Zbl 1439.20051

Summary: A group $$G$$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group $$G$$ has a generating set $$X$$ such that the corresponding Cayley graph $$\Gamma$$ is a (non-elementary) quasi-tree and the action of $$G$$ on $$\Gamma$$ is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As an application, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.

### MSC:

 20F67 Hyperbolic groups and nonpositively curved groups 20F65 Geometric group theory 20E08 Groups acting on trees 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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### References:

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