Götz Gelbrich:
Self-similar periodic tilings on the Heisenberg group  
Journal of Lie Theory, vol. 4 (1), p. 31-37

We construct a tiling on the Heisenberg group $G$with the
following properties. A discrete cocompact subgroup of 
$G$acts freely and
transitively on the set of tiles. Moreover, an expanding endomorphism
of $G$carries each tile onto the union of 
$k$tiles, where $k=4$, and
this is the least number for which such a construction is possible.
Our computations are basic for the generation of arbitrary periodic
self-similar tilings on $G$.