Tiling Spaces have aspherical shape
Scott Schmieding

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 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.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Geometric and Algebraic Topology · Geometric Analysis and Curvature Flows
