Some applications of algebraic entropy to the proof of Milnor-Wolf theorem

We discuss the growth function of a finitely generated cascade and its connection to the growth function of its related semi-direct product (Conjecture 1.9). The results is applied for simpler proof of well-known results in the realm of geometric group theory. We show that the finitely generated cascades on nilpotent groups obey the dichotomy rule (only polynomial and exponential growth are possible).