Let $X$ be a proper geodesic Gromov hyperbolic space whose isometry group contains a uniform lattice $\Gamma$. For instance, $X$ could be a negatively curved contractible manifold or a Cayley graph of a hyperbolic group. Let $H$ be a discrete subgroup of isometries of $X$ with critical exponent (exponential growth rate) strictly less than half of the growth rate of $\Gamma$. We show that the injectivity radius of $X/H$ grows linearly along almost every geodesic in $X$ (with respect to the Patterson-Sullivan measure on the Gromov boundary of $X$). The proof will involve an elementary analysis of a novel concept called the "sublinearly horospherical limit set" of $H$ which is a generalization of the classical concept of "horospherical limit set" for Kleinian groups. This talk is based on joint work with Inhyeok Choi and Keivan Mallahi-Kerai.