  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.
