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Acylindrically hyperbolic groups


Author: D. Osin
Journal: Trans. Amer. Math. Soc. 368 (2016), 851-888
MSC (2010): Primary 20F67, 20F65
DOI: https://doi.org/10.1090/tran/6343
Published electronically: May 22, 2015
MathSciNet review: 3430352
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Abstract: We say that a group $ G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class $ C_{geom}$ introduced by Hamenstädt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic space in the sense of Bestvina and Fujiwara, and the class of groups with hyperbolically embedded subgroups studied by Dahmani, Guirardel, and the author. We also record some basic results about acylindrically hyperbolic groups for future use.


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Additional Information

D. Osin
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: denis.v.osin@vanderbilt.edu

DOI: https://doi.org/10.1090/tran/6343
Received by editor(s): April 8, 2013
Received by editor(s) in revised form: November 8, 2013, and December 4, 2013
Published electronically: May 22, 2015
Additional Notes: This work was supported by NSF grant DMS-1006345 and by RFBR grant 11-01-00945
Article copyright: © Copyright 2015 American Mathematical Society