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 $Ω$ can be written as an inverse limit $\varprojlim Γ_n$, then the étale fundamental group of $Ω$, which is defined via a limit of covers, is isomorphic to the inverse limit $\hat π_1(Ω) := \varprojlim \hat π_1(Γ_n)$ of the profinite completions of the fundamental groups $π_1(Γ_n)$. This isomorphism allows us to construct all covers of tiling spaces and to use those covers to distinguish spaces that have identical cohomology groups.