The notion of topological predictability was introduced by
Kamiński, Siemaszko and Szymański in 2003. A topological dynamical
system $(X,T)$ is said to be topologically predictable if every
continuous function on $X$ belongs to the closed algebra generated by
$1,Tf,T^2f,T^3f,\ldots$. Kamiński, Siemaszko and Szymański showed in 2005
that topologically predictable systems have zero topological entropy. In
2012, Hochman extended this result to $\mathbb{Z}^d$-actions, and asked
if it also holds for amenable group actions. In this talk, we partially
answer this question affirmatively for actions of torsion-free nilpotent
groups. A key tool is an algebraic theorem due to Rhemtulla and
Formanek. We will give an independent proof of the Rhemtulla-Formanek
theorem for torsion-free abelian groups and the Heisenberg group. This
is a joint work with Huang and Ye.