To every countable amenable group G and every element f of its
integral group ring ZG, there is associated via Pontryagin duality for
ZG/ZGf a very interesting action of G by automorphisms of the compact
abelian dual of ZG/ZGf. For abelian groups G, natural dynamical
questions such as expansiveness and entropy have been more or less
completely worked out, involving ideas such as varieties, amoebas, and
the Mahler measure of polynomials.
Recent work of Denninger suggest an approach to these problems for
noncommutative G, in particular the integral Heisenberg group, using von
Neumann algebra theory. For entropy, there appears to be a concrete
description in terms of a new kind of Mahler measure, for polynomials in
noncommuting variables, which uses the multiplicative ergodic theorem
for its computation.
I’ll start with some background using examples, and then discuss actions
of the integral Heisenberg group and the many open problems these
suggest.