On Growth of Double Cosets in Hyperbolic Groups

TL;DR

This paper investigates the growth of double cosets in hyperbolic groups, revealing a dichotomy: non-quasiconvex subgroups can produce arbitrary growth functions, while quasiconvex subgroups exhibit exponential growth with a lower bound proportional to the group’s growth.

Contribution

It establishes a clear dichotomy in the growth behavior of double cosets based on subgroup quasiconvexity in hyperbolic groups, including the realization of arbitrary growth functions.

Findings

Non-quasiconvex subgroups can produce any finitely presented group growth function.

Quasiconvex subgroups of infinite index have exponential growth of double cosets.

A lower bound proportional to the group's growth function exists for quasiconvex subgroups.

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 $AhB, h \in H$. 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 $\lambda > 0$, such that $gr(H,A,B)(r) >\lambda f_H(r)$ 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.