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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
Article
  • 38 Downloads

Abstract

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, \(\langle H_i, \gamma \rangle = H_i * \langle \gamma \rangle \).

Keywords

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

Mathematics Subject Classification

20E07 20F65 46L35 

Notes

Acknowledgements

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.

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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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