Simple Tilings of Nilpotent Lie Groups

We define simple tilings in the general context of a $G$-tiling on a Riemannian homogeneous space $M$ to be tilings by Riemannian simplices. As evidence that this definition is natural, we prove that a large class of tilings of $M$ are MLD to simple ones. We demonstrate the utility of this definition by generalizing previously known results about simple tilings of Euclidean space. In particular, it is shown that a simple tiling space of a rational, connected, simply connected, nilpotent Lie group is homeomorphic to a rational tiling space, that is, a tiling space for which displacement between vertices take on rational values. Hence, such a tiling space is a fiber bundle over a nilmanifold. We further sketch a proof of the fact that there is an isomorphism between \v{C}ech cohomology and pattern equivariant cohomology of simple tilings in connected, simply connected, nilpotent Lie groups.