Geodesic
BiLipschitz homogeneous hyperbolic nets
Christopher J. Bishop1
1 Stony Brook University, Mathematics Department
Annales Fennici Mathematici, Tome 49 (2024) no. 2, p. 685–694.

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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.
DOI : 10.54330/afm.152404
Keywords: BiLipschitz maps, hyperbolic geometry, Margulis constant, homogeneous set, quadrilateral mesh

Christopher J. Bishop 1

1 Stony Brook University, Mathematics Department 
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Christopher J. Bishop. BiLipschitz homogeneous hyperbolic nets. Annales Fennici Mathematici, Tome 49 (2024) no. 2, p. 685–694. doi : 10.54330/afm.152404. http://geodesic.mathdoc.fr/articles/10.54330/afm.152404/

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