Covers of Tiling Spaces

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 $\mathrm{\Omega }$ can be written as an inverse limit $\mathrm{}{\mathrm{\Gamma }}_n$, then the étale fundamental group of $\mathrm{\Omega }$, which is defined via a limit of covers, is isomorphic to the inverse limit ${\widehat{\pi }}_1(\mathrm{\Omega }):=\mathrm{}{\widehat{\pi }}_1({\mathrm{\Gamma }}_n)$ of the profinite completions of the fundamental groups ${\pi }_1({\mathrm{\Gamma }}_n)$. This isomorphism allows us to construct all covers of tiling spaces and to use those covers to distinguish spaces that have identical cohomology groups.