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 language | English |
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Pages (from-to) | 1161-1166 |
Number of pages | 6 |
Journal | International Journal of Algebra and Computation |
Volume | 30 |
Issue number | 6 |
DOIs | |
State | Published - 1 Sep 2020 |
Bibliographical note
Publisher Copyright:© 2020 World Scientific Publishing Company.
Keywords
- double coset
- Growth function
- hyperbolic group
- quasiconvex subgroup