Growth of quasi-convex subgroups in groups with a constricting element

TL;DR

This paper investigates the growth rates of quasi-convex subgroups in groups acting on metric spaces, especially when a constricting element exists, with applications to various geometric group classes.

Contribution

It characterizes conditions under which relative and quotient exponential growth rates of quasi-convex subgroups differ or match the group's growth rate, extending understanding in geometric group theory.

Findings

Identifies when relative exponential growth rates are strictly smaller than the group's growth rate.

Determines when quotient exponential growth rates coincide with the group's growth rate.

Applies results to relatively hyperbolic, CAT(0), and hierarchically hyperbolic groups with Morse elements.

Abstract

Given a group G acting on a geodesic metric space, we consider a preferred collection of paths of the space -- a path system -- and study the spectrum of relative exponential growth rates and quotient exponential growth rates of the infinite index subgroups of G which are quasi-convex with respect to this path system. If G contains a constricting element with respect to the same path system, we are able to determine when the first kind of growth rates are strictly smaller than the growth rate of G, and when the second kind of growth rates coincide with the growth rate of G. Examples of applications include relatively hyperbolic groups, CAT(0) groups and hierarchically hyperbolic groups containing a Morse element.