The theory of actions of Z^d by automorphisms of a compact abelian group has been extensively worked out over the past twenty years, with fascinating connections with commutative algebra, algebraic geometry, and tropical geometry. The same formal framework extends to defining algebraic actions of general discrete groups such as the discrete Heisenberg group. However questions which have good answers in the commutative case are now more "interesting"! I'll discuss recent work with Schmidt on the problem of computing entropy for such actions, leading to a natural definition of Mahler measure for polynomials in noncommuting variables, which appears to be a new idea.