Classification of tiling $C^*$-algebras

Abstract

We prove that Kellendonk's $C^*$-algebra of an aperiodic and repetitive tiling with finite local complexity is classifiable by the Elliott invariant. Our result follows from showing that tiling $C^*$-algebras are $\mathcal{Z}$-stable, and hence have finite nuclear dimension. To prove $\mathcal{Z}$-stability, we extend Matui's notion of almost finiteness to the setting of \'etale groupoid actions following the footsteps of Kerr. To use some of Kerr's techniques we have developed a version of the Ornstein-Weiss quasitiling theorem for general \'etale groupoids.