Résumé
Let$\Gamma'<\Gamma$be two discrete groups acting properly by isometries on a Gromov-hyperbolic space$X$ . We prove that their critical exponents coincide if and only if$\Gamma'$is co-amenable in$\Gamma$ , under the assumption that the action of$\Gamma$on$X$is strongly positively recurrent, i.e. has a growth gap at infinity. This generalizes all previously known results on this question, which required either$X$to be the real hyperbolic space and$\Gamma$geometrically finite, or$X$Gromov hyperbolic and$\Gamma$cocompact. This result is optimal: we provide several counterexamples when the action is not strongly positively recurrent.