Covers of Tiling Spaces

Abstract

We study the ways that one tiling space can be a finite regular cover of another. We classify all of the finite regular covers of a tiling space via its structure as an inverse limit space. If the tiling space $\hat{I}\text{c◯}$ can be written as an inverse limit $\underleftarrow{\lim }\hat{I}{“}_n$, then the étale fundamental group of $\hat{I}\text{c◯}$, which is defined via a limit of covers, is isomorphic to the inverse limit $\hat{\ddot{I}}{€}_1(\hat{I}\text{c◯}):=\underleftarrow{\lim }\hat{\ddot{I}}{€}_1(\hat{I}{“}_n)$ of the profinite completions of the fundamental groups $\ddot{I}{€}_1(\hat{I}{“}_n)$. This isomorphism allows us to construct all covers of tiling spaces and to use those covers to distinguish spaces that have identical cohomology groups.