Algebraic actions of the discrete Heisenberg group

Let G be a countable discrete group, and let R = Z[G]
be its integral group ring. The natural G-action on R/Rf
dualizes to an interesting G-action on the compact abelian
dual group of R/Rf. For G the integers, or generally the d-dimensional
integers, there is a beautiful connection relating dynamical
questions to algebra and number theory. I’ll briefly describe
some aspects of this, and then discuss ongoing work with Klaus
Schmidt for the case when G is the discrete Heisenberg group.
Here there are only partial answers and many open questions.