On Extensions over Semigroups and Applications

Applying a theorem according to Rhemtulla and Formanek, we partially solve an open problem raised by Hochman with an affirmative answer. Namely, we show that if G is a countable torsion-free locally nilpotent group that acts by homeomorphisms on X, and S ⊂ G is a subsemigroup not containing the unit of G such that f ∈ 〈1, s f : s ∈ S〉 for every f ∈ C(X), then (X, G) has zero topological entropy.