C*-algebras of Penrose's hyperbolic tilings

TL;DR

This paper provides a comprehensive analysis of the C*-algebras associated with Penrose hyperbolic tilings, including their K-theory and gap-labelling, despite the absence of invariant measures.

Contribution

It offers a complete description of the C*-algebras and K-theory for Penrose hyperbolic tilings, introducing higher-order gap-labelling via harmonic currents.

Findings

C*-algebras are traceless due to lack of invariant measures

K-theory for these tilings is explicitly characterized

Higher-order gap-labelling is developed using harmonic currents

Abstract

Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit, up to affine transformations a finite number of prototiles. In this paper, we give a complete description of the C*-algebras and of the K-theory for such tilings. Since the continuous hull of these tilings have no transversally invariant measure, these C*-algebras are traceless. Nevertheless, harmonic currents give rise to 3-cyclic cocycles and we discuss in this setting a higher-order version of the gap-labelling.