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.