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

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, CAT(0) groups, and hierarchically hyperbolic groups containing a Morse element.