Tiling spaces are inverse limits

Let M be an arbitrary Riemannian homogeneous space, and let Ω be a space of tilings of M, with finite local complexity (relative to some symmetry group Γ) and closed in the natural topology. Then Ω is the inverse limit of a sequence of compact finite-dimensional branched manifolds. The branched manifolds are (finite) unions of cells, constructed from the tiles themselves and the group Γ. This result extends previous results of Anderson and Putnam, of Ormes, Radin, and Sadun, of Bellissard, Benedetti, and Gambaudo, and of Gahler. In particular, the construction in this paper is a natural generalization of Gahler’s.