K-theory of two-dimensional substitution tiling spaces from AF-algebras

Abstract

Given a two-dimensional substitution tiling space, we show that, under some reasonable assumptions, the K-theory of the groupoid C*-algebra of its unstable groupoid can be explicitly reconstructed from the K-theory of the AF-algebras of the substitution rule and its analogue on the 1-skeleton. We prove this by generalizing the calculations done for the chair tiling in [JS16] using relative K-theory and excision, and packaging the result into an exact sequence purely in topology. From this exact sequence, it appears that one cannot use only ordinary K-theory to compute using the dimension-filtration on the unstable groupoid. Several examples are computed using Sage and the results are compiled in a table.