Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 555–568 | Cite as

Property \(P_{naive}\) for acylindrically hyperbolic groups

  • Carolyn R. AbbottEmail author
  • François Dahmani


We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the \(P_{naive}\) property: for any finite collection of elements \(h_1, \dots , h_k\), there exists another element \(\gamma \ne 1\) such that for all i, \(\langle h_i, \gamma \rangle = \langle h_i \rangle * \langle \gamma \rangle \). We also show that if a collection of subgroups \(H_1, \dots , H_k\) is a hyperbolically embedded collection, then there is \(\gamma \ne 1\) such that for all i, \(\langle H_i, \gamma \rangle = H_i * \langle \gamma \rangle \).


Acylindrically hyperbolic groups \(\delta \)-hyperbolic spaces \(C^*\)-algebras Free subgroups Ping-pong lemma Property \(P_{naive}\) 

Mathematics Subject Classification

20E07 20F65 46L35 



The authors thank M. Sageev for asking the question and encouraging us to write this note. The authors also thank D. Osin for pointing out the application to group rings, as well as the anonymous referee for useful comments. C.A. is partially supported by NSF RTG award DMS-1502553. F.D. is supported by the Institut Universitaire de France. The authors would like to thank the Mathematical Sciences Research Institute for hosting them during the completion of this project.


  1. 1.
    Alexander, J., Nishinaka, T.: Non-noetherian groups and primitivity of their group algebras. J. Algebra 473, 221–246 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arzhantseva, G., Minasyan, A.: Relatively hyperbolic groups are C*-simple. J. Funct. Anal. 243(1), 345–351 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balogun, B.O.: On the primitivity of group rings of amalgamated products. Proc. Am. Math. Soc. 106, 43–47 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bekka, M., Cowling, M., de la Harpe, P.: Some groups whose reduced \(C^*\)-algebra is simple. Inst. Hautes Études Sci. Publ. Math. 80, 117–134 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bestvina, M., Bromberg, K., Fujiwara, K.: Constructing group actions on quasi-trees and applications to mapping class groups. Publ. Math. IHES 122, 1 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bowditch, B.: Tight geodesics in the curve complex. Invent. Math. 171(2), 105–129 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Breuillard, E., Kalantar, M., Kennedy, M., Ozawa, N.: C*-simplicity and the unique trace property for discrete groups. Publ. Math. Inst. Hautes Étud. Sci. 126, 35–71 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahmani, F., Guirardel, V., Osin, D.: Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Memoirs Am. Math. Soc. 245(1156), v+152 (2017)
  9. 9.
    Formanek, E.: Group rings of free products are infinite. J. Algebra 26, 508–511 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Formanek, E., Snider, R.L.: Primitive group rings. Proc. Am. Math. Soc. 36, 357–360 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hamenstädt, U.: Bounded cohomology and isometry groups of hyperbolic spaces. J. Eur. Math. Soc. 10(2), 315–349 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    de la Harpe, P.: Groupes hyperboliques, algèbres d’opérateurs et un théorème de Jolissaint. C. R. Acad. Sci. Paris Sér. I Math. 307(14), 771–774 (1988)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kalantar, M., Kennedy, M.: Boundaries of reduced \(C^*\)—algebras of discrete groups. J. Reine Angew. Math. 727, 247–267 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kar, A., Sageev, M.: Ping Pong on CAT(0) cube complexes. Comment. Math. Helv. 91(3), 543–561 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Osin, D.: Acylindrically hyperbolic groups. Trans. Am. Math. Soc. 368(2), 851–888 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Solie, B.: Primitivity of group rings of non-elementary torsion-free hyperbolic groups. J. Algebra 493, 438–443 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA
  2. 2.Université Grenoble AlpesInstitut FourierGrenobleFrance

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