Tessellations of solvmanifolds

Let $A$ be a closed subgroup of a connected, solvable Lie group $G$, such that the homogeneous space $A\backslash G$ is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation $A\backslash G/\Gamma$ of $A\backslash G$ is finitely covered by a compact homogeneous space $G'/\Gamma'$. We prove that the covering map can be taken to be very well behaved - a ``crossed" affine map. This establishes a connection between the geometry of the tessellation and the geometry of the homogeneous space. In particular, we see that every geometrically-defined flow on $A\backslash G/\Gamma$ that has a dense orbit is covered by a natural flow on $G'/\Gamma'$.