Gromov asked: suppose G is a f.g group that contains a free group, does there exist an integer m such that for any​ finite generating set A, there is a pair of elements g,h in G in a ball of radius m in Cay(G,A) generating a free group?

More generally, suppose that G is a group that contains a subgroup H<G with some interesting dynamics, does there exist an integer m such that for any finite generating set A, the ball of radius m about the identity contains elements h_1,..,h_k generating a subgroup H=<h_1,...,h_k> with the desired interesting dynamics?

A subgroup H<G is said to be stable if it is both hyperbolic, and quasi-geodesics with end points on H remain close to H (these characterize convex-cocompact subgroups of mapping class groups and purely loxodromic subgroups of RAAGS). I will show that for a large class of groups, if convex cocompact subgroups exist, then they can be found uniformly quickly in the above sense.