Formula omitted]-algebras of Penrose hyperbolic tilings

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-labeling.