Critical exponents for normal subgroups and spectral gaps

Let $X$ be a simply connected, complete Riemannian manifold with pinched negative sectional curvatures. The critical exponent $\delta\Gamma$ of a discrete group of isometries $\Gamma$ of $X$ is equal to the abscissa of convergence of the Poincaré series of $\Gamma$; and for a cocompact $\Gamma0$, this is given by the volume growth of $X$. (Moreover, if $X$ is a symmetric space then there is a relationship between $\delta\Gamma$ and the bottom of the spectrum of the Laplacian on $X/\Gamma$.) Using a more dynamical approach, we characterise the existence of a uniform gap $\delta\Gamma<\delta{\Gamma0}$ for a family of (infinite index) normal subgroups $\Gamma$ of $\Gamma0$, in terms of permutation representations given by the quotients $\Gamma0/\Gamma$.