On growth of double cosets in hyperbolic groups

Rita Gitik, Eliyahu Rips

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let H be a hyperbolic group, A and B be subgroups of H, and gr(H,A,B) be the growth function of the double cosets AhB,h H. We prove that the behavior of gr(H,A,B) splits into two different cases. If A and B are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as gr(H,A,B). We can even take A = B. In contrast, for quasiconvex subgroups A and B of infinite index, gr(H,A,B) is exponential. Moreover, there exists a constant λ > 0, such that gr(H,A,B)(r) > λfH(r) for all big enough r, where fH(r) is the growth function of the group H. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

Original languageEnglish
Pages (from-to)1161-1166
Number of pages6
JournalInternational Journal of Algebra and Computation
Volume30
Issue number6
DOIs
StatePublished - 1 Sep 2020

Bibliographical note

Publisher Copyright:
© 2020 World Scientific Publishing Company.

Keywords

  • double coset
  • Growth function
  • hyperbolic group
  • quasiconvex subgroup

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