Tiling Spaces have aspherical shape

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.