Growth of quasiconvex subgroups

We prove that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The proof uses the classical tools of automatic structures and Perron–Frobenius theory.We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel and Osin on rotating families.