BiLipschitz homogeneous hyperbolic nets

Christopher J. Bishop¹

¹Stony Brook University, Mathematics Department

Annales Fennici Mathematici, Tome 49 (2024) no. 2, p. 685–694

Résumé

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.

Détail

DOI : 10.54330/afm.152404

Keywords: BiLipschitz maps, hyperbolic geometry, Margulis constant, homogeneous set, quadrilateral mesh