Let γ be a geodesic in a Gromov hyperbolic metric space (such as hyperbolic space itself). Then γ satisfies several very useful "attracting" properties, which morally, all state that there are no "short-cuts" avoiding γ: any efficient path with end points on gamma must stay relatively close by to γ. We prove that several of these properties are actually equivalent. As an application, we give new ways of detecting when a subgroup H of a finitely generated group G is "stable", which is a strong version of quasiconvexity. We then use these criteria to prove that stability is generic in two important cases: a random finitely generated subgroup of either the mapping class group or the outer automorphism group of the free group is stable. This represents joint work with Matthew Durham and Samuel Taylor.