C∗-algebras of Penrose hyperbolic tilings

Abstract

a b s t r a c tPenrose 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.

K-theory of the chair tiling via AF-algebras

“…We consider proto-edges modulo rotations ρ = e ıkπ , k = 0, 1. We set a = [1 a ], b = [1 b ], c = [2], d = [5], e = [6], and specify edges with a given orientation:…”

“…1 In particular, the ordered K 0 -group of this C * -algebra can be related to the gaps in the spectrum of a Schrödinger operator describing the motion of electrons on a quasicrystal [2]. There have been many approaches for computing topological invariants of tiling spaces and their C * -algebras, and we very partially decide to cite Kellendonk [3] (one of the first approaches, relating K -theory to the group of coinvariants), Moustafa [4] (K -theory computations for the Pinwheel tiling, involving the explicit construction of fiber bundles representing K -elements), and Oyono-Oyono-Petite [5] (very sophisticated computations involving the K -theory of the hyperbolic Penrose tiling). Beyond particular examples, we can single out two important families of aperiodic, repetitive tilings: self-similar tilings and cutand-project tilings (also known as model sets).…”

On the Noncommutative Geometry of Tilings

“…Newer developments concern the noncommutative topology of tilings with infinite rotational symmetry, like the Pinwheel tilings [45], of tilings in hyperbolic space [46] and of combinatorial tilings [50,51]. These developments will not be explained in this book.…”

K-theory of the Chair Tiling via AF-algebras

“…1 In particular, the ordered K 0 -group of this C * -algebra can be related to the gaps in the spectrum of a Schrödinger operator describing the motion of electrons on a quasicrystal [4]. There have been many approaches for computing topological invariants of tiling spaces and their C * -algebras, and we very partially decide to cite Kellendonk [11] (one of the first approaches, relating K-theory to the group of coinvariants), Moustafa [13] (K-theory computations for the Pinwheel tiling, involving the explicit construction of fiber bundles representing K-elements), and Oyono-Oyono-Petite [15] (very sophisticated computations involving the K-theory of the hyperbolic Penrose tiling). Beyond particular examples, we can single out two important families of aperiodic, repetitive tilings: self-similar tilings and cut-andproject tilings (also known as model sets).…”