Cut and Project Schemes in the Poincaré Disc: From Cocompact Fuchsian Groups to Chaotic Delone Sets

This paper addresses a question concerning the development of new cut and project models for generating better-performing graded metamaterials. It studies a natural construction of such a scheme in relation to cocompact Fuchsian groups acting on the Poincaré disc model of hyperbolic space. The authors present a condition on the fundamental domain of the group that ensures the resulting cut and project set is a chaotic Delone set. The research also investigates the set of tile lengths, proving it to be countably infinite. Finally, the results are applied to cocompact Fuchsian triangle groups, showing that the resulting cut and project sets are chaotic Delone. This work is significant for the design and understanding of metamaterials and quasicrystals.