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.


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.)
Full Text: DOI arXiv


[1] 10.1007/s10240-014-0067-4 · Zbl 1372.20029
[2] 10.1515/CRELLE.2006.070 · Zbl 1119.32006
[3] 10.1007/978-3-662-12494-9
[4] 10.1090/memo/1156 · Zbl 1396.20041
[5] 10.1112/jtopol/jtt027 · Zbl 1348.20044
[6] 10.4171/GGD/231 · Zbl 1315.20022
[7] 10.1142/S021819671450009X · Zbl 1342.20042
[8] 10.2140/gt.2005.9.1147 · Zbl 1083.20038
[9] 10.1112/blms/bdv047 · Zbl 1345.20059
[10] 10.1090/tran/6343 · Zbl 1380.20048
[11] 10.1112/blms/14.1.45 · Zbl 0481.20020
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