C*-algebras of Penrose's hyperbolic tilings

Abstract

36 pages. v2: some mistakes corrected, a section on topological invariants of the continuous hull of the Penrose hyperbolic tilings added.International audiencePenrose 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