Let
A
A
be a closed subgroup of a connected, solvable Lie group
G
G
, such that the homogeneous space
A
∖
G
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
∖
G
/
Γ
A\backslash G/\Gamma
of
A
∖
G
A\backslash G
is finitely covered by a compact homogeneous space
G
′
/
Γ
′
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
∖
G
/
Γ
A\backslash G/\Gamma
that has a dense orbit is covered by a natural flow on
G
′
/
Γ
′
G’/\Gamma ’
.