BiLipschitz homogeneous hyperbolic nets

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We answer a question of Itai Benjamini by showing there is a \(K< \infty\) so that for any \(\epsilon >0\), there exist \(\epsilon\)-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 \(\epsilon(K)>0\). For \(K=1\), this is a consequence of the Margulis lemma for discrete groups of hyperbolic isometries.