On growth of double cosets in hyperbolic groups

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.