Cut and project schemes in the Poincaré disc: From cocompact Fuchsian groups to chaotic Delone sets

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 Poincaré 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 ${\mathcal{L}}_S=\{z-y:z,y\in S,z>y\text{and}(y,z)\cap S=\mathrm{\varnothing }\}$, 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 López et al. [Discrete Contin. Dyn. Syst. 41, 2021].