On Growth of Double Cosets in Hyperbolic Groups

Rita Gitik; Eliyahu Rips

paper

On Growth of Double Cosets in Hyperbolic Groups

arXiv:1808.00802

Abstract

Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$ , and $g r (H, A, B)$ be the growth function of the double cosets $A h B, h \in H$ . We prove that the behavior of $g r (H, A, B)$ 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 $g r (H, A, B)$ . We can even take $A = B$ . In contrast, for quasiconvex subgroups A and B of infinite index, $g r (H, A, B)$ is exponential. Moreover, there exists a constant $λ > 0$ , such that $g r (H, A, B) (r) > λ f_{H} (r)$ for all big enough $r$ , where $f_{H} (r)$ is the growth function of the group $H$ . So, we have a clear dychotomy between the quasiconvex and non-quasiconvex case.