Algebraic & Geometric Topology

Acylindrical group actions on quasi-trees

Sahana Balasubramanya

Abstract

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

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

Dates
Revised: 19 October 2016
Accepted: 30 October 2016
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510841440

Digital Object Identifier
doi:10.2140/agt.2017.17.2145

Mathematical Reviews number (MathSciNet)
MR3685605

Zentralblatt MATH identifier
06762688

Citation

Balasubramanya, Sahana. Acylindrical group actions on quasi-trees. Algebr. Geom. Topol. 17 (2017), no. 4, 2145--2176. doi:10.2140/agt.2017.17.2145. https://projecteuclid.org/euclid.agt/1510841440

References

• J Behrstock, M,F Hagen, A Sisto, Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups, preprint (2015)
• M Bestvina, K Bromberg, K Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015) 1–64
• B,H Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006) 105–129
• M,R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer (1999)
• F Dahmani, V Guirardel, D Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 1156, Amer. Math. Soc., Providence, RI (2017)
• M,F Hagen, Weak hyperbolicity of cube complexes and quasi-arboreal groups, J. Topol. 7 (2014) 385–418
• I Kapovich, K Rafi, On hyperbolicity of free splitting and free factor complexes, Groups Geom. Dyn. 8 (2014) 391–414
• S-H Kim, T Koberda, The geometry of the curve graph of a right-angled Artin group, Internat. J. Algebra Comput. 24 (2014) 121–169
• J,F Manning, Geometry of pseudocharacters, Geom. Topol. 9 (2005) 1147–1185
• A Minasyan, D Osin, Small subgroups of acylindrically hyperbolic groups, preprint (2016)
• D Osin, On acylindrical hyperbolicity of groups with positive first $\ell^2$–Betti number, Bull. Lond. Math. Soc. 47 (2015) 725–730
• D Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016) 851–888
• E Rips, Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982) 45–47
• A Sisto, On metric relative hyperbolicity, preprint (2012)