  The study of actions of countable groups by automorphisms of compact abelian
groups has recently undergone intensive development, revealing deep connections
with operator algebras and other areas. The discrete Heisenberg group is the
simplest noncommutative example, where dynamical phenomena related to its
noncommutativity already illustrate many of these connections. The explicit
structure of this group means that these phenomena have concrete descriptions,
which are not only instances of the general theory but are also testing grounds
for further work. We survey here what is known about such actions of the
discrete Heisenberg group, providing numerous examples and emphasizing many of
the open problems that remain.
