On $K_0$-Groups for Substitution Tilings

TL;DR

This paper investigates the structure of $K_0$-groups associated with substitution tilings, providing explicit calculations for Penrose tilings and linking tiling invariants to algebraic $K$-theory.

Contribution

It determines the $K_0$-groups for a class of substitution tilings, including Penrose tilings, using group-theoretic methods and duality techniques.

Findings

The group $C( ext{Omega}, ext{Z})/E$ is explicitly computed for certain tilings.

For Penrose tilings, the $K_0$-group is $ ext{Z}^8 imes ext{Z}$.

The $K_0$-group relates to the algebraic structure of the tiling space.

Abstract

The group $C(\Om,\Z)/\E$ is determined for tilings which are invariant under a locally invertible primitive \sst\ which forces its \saum. In case the tiling may be obtained by the generalized dual method from a regular grid this group furnishes part of the $K_0$-group of the algebra of the tiling. Applied to Penrose tilings one obtains $K_0(\A_\tl)=\Z^8\oplus\Z$.