BiLipschitz homogeneous hyperbolic nets

Abstract

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