  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.
