The Local Structure of Tilings and their Integer Group of Coinvariants

TL;DR

This paper explores the local structure of tilings through a multiplicative pattern class framework, linking it to groupoid $C^*$-algebras and $K$-theory, with applications to Schrödinger operators.

Contribution

It introduces a new multiplicative structure on tiling pattern classes and computes the associated integer group of coinvariants for substitution tilings.

Findings

Computed the integer group of coinvariants for certain substitution tilings.

Connected the group of coinvariants to the $K_0$-group of the tiling's groupoid $C^*$-algebra.

Predicted gap labels for Schrödinger operators based on $K$-theory.

Abstract

The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the $K_0$-group of the groupoid $C^*$-algebra for tilings which reduce to decorations of $\Z^d$. The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and primitive. This yields in particular the set of possible gap labels predicted by $K$-theory for Schr\"odinger operators describing the particle motion in such a tiling.