Tessellations of Solvmanifolds

Let A be a closed subgroup of a connected, solvable Lie group G, such that the homogeneous space A\G is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation A\G/Γ of A\G is finitely covered by a compact homogeneous space G′/Γ′. 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\G/Γ that has a dense orbit is covered by a natural flow on G′/Γ′.