$K$-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 $C^\ast$-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.