BiLipschitz homogeneous hyperbolic nets

We answer a question of Itai Benjamini by showing there is a K < ∞ so that for any ǫ?> 0, there exist ǫ-dense discrete sets in the hyperbolic disk that are homogeneous with respect to K-biLipschitz maps of the disk to itself. However, this is not true for K close to 1; in that case, every K-biLipschitz homogeneous discrete set must omit a disk of hyperbolic radius ǫ(K)?> 0. For K = 1, this is a consequence of the Margulis lemma for discrete groups of hyperbolic isometries.