Cut and project schemes in the Poincar\'e disc: From cocompact Fuchsian groups to chaotic Delone sets

TL;DR

This paper explores a novel hyperbolic geometric construction of cut and project schemes related to cocompact Fuchsian groups, producing chaotic Delone sets with infinitely many tile lengths, extending previous work in hyperbolic tilings.

Contribution

It introduces a new method for constructing chaotic Delone sets via cut and project schemes in hyperbolic space, specifically using cocompact Fuchsian groups and hyperbolic polygons.

Findings

Chaotic Delone sets can be generated from hyperbolic Fuchsian groups.

The set of tile lengths in these sets is countably infinite.

Application to cocompact Fuchsian triangle groups extends prior hyperbolic tiling results.

Abstract

A question raised by Davies et al [Phys. Rev. Lett. 131, 2023] is: "Can developing new cut and project models, where the lattice is not square or the curve is non-linear, generate better performing graded metamaterials?" In this article, we study a natural construction of such a cut and project scheme, namely, cut and project schemes in relation to cocompact Fuchsian groups acting on the Poincar\'e disc model of hyperbolic space. We present a condition on the fundamental domain (a hyperbolic polygon) of the group so that the resulting cut and project set $S \subset \mathbb{R}$ is a chaotic Delone set. We also investigate the set of tile lengths of $S$, namely $\mathcal{L}_{S} = \{ z - y : z,y \in S, \, z > y \; \text{and} \; (y,z) \cap S = \emptyset \}$, and show that this set is countably infinite. Finally, we apply our results to cocompact Fuchsian triangle groups and show that the resulting cut and project sets are chaotic Delone, complementing and extending the work of L\'{o}pez et al. [Discrete Contin. Dyn. Syst. 41, 2021].