Algebraic & Geometric Topology

Acylindrical group actions on quasi-trees

Sahana Balasubramanya

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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 Γ is a (non-elementary) quasi-tree and the action of G on Γ 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.

Article information

Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2145-2176.

Received: 4 March 2016
Revised: 19 October 2016
Accepted: 30 October 2016
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20E08: Groups acting on trees [See also 20F65]

acylindrically hyperbolic groups acylindrical actions projection complex quasi-trees hyperbolically embedded subgroups


Balasubramanya, Sahana. Acylindrical group actions on quasi-trees. Algebr. Geom. Topol. 17 (2017), no. 4, 2145--2176. doi:10.2140/agt.2017.17.2145.

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