Tiling Spaces have aspherical shape

TL;DR

This paper investigates the shape invariants of tiling spaces, demonstrating that higher-dimensional shape groups are trivial by linking tiling hulls to inverse limits of aspherical manifolds and applying geometric group theory results.

Contribution

It extends the understanding of shape invariants in tiling spaces by proving triviality of higher-dimensional shape groups using geometric and topological methods.

Findings

Shape groups in dimensions greater than one are trivial.

Tiling hulls are inverse limits of aspherical branched manifolds.

Utilizes Gromov's work on nonpositive curvature spaces.

Abstract

Going beyond the cohomological invariants attached to tiling spaces via inverse limit constructions, Clark and Hunton introduced shape group invariants, and showed these invariants in dimension one give new information. We show for dimensions greater than one that these groups are trivial. To do this, we show that the topological hull associated to a tiling in $\mathbb{R}^{d}$ is the inverse limit of aspherical branched manifolds. The result follows from work of Gromov and others on complete geodesic spaces of nonpositive Alexandrov curvature, along with some simple observations.