Twisted Patterson-Sullivan measures and applications to amenability and coverings

Abstract

Let ${\Gamma }^{\prime }<\Gamma$ 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 ${\Gamma }^{\prime }$ 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.