In groups containing a constricting element w.r.t. a path system, relative exponential growth rates of infinite-index quasi-convex subgroups are strictly smaller than the group growth rate, while quotient exponential growth rates coincide with it.
Regularity of morse geodesics and growth of stable subgroups
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Growth of quasi-convex subgroups in groups with a constricting element
In groups containing a constricting element w.r.t. a path system, relative exponential growth rates of infinite-index quasi-convex subgroups are strictly smaller than the group growth rate, while quotient exponential growth rates coincide with it.