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

Abstract

Given a group $G$ acting by isometries on a metric space $X$, we consider a preferred collection of paths of the space $X$, a path system, and study the spectrum of relative exponential growth rates and quotient exponential growth rates of the infinite index subgroups of $G$ that 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 growth rates of the first kind are strictly smaller than the growth rate of $G$, and when the growth rates of the second kind coincide with the growth rate of $G$. Examples of applications include relatively hyperbolic groups, $\mathrm{CAT}(0)$ groups, and hierarchically hyperbolic groups containing a Morse element.