BiLipschitz homogeneous hyperbolic nets
Year: 2024
Type: article
Abstract: 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.
Source: Annales Fennici Mathematici
DOI: 10.54330/afm.152404
Author: Christopher J. Bishop
Institution: Stony Brook University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404734761