Nilpotent torsion-free Groups with a fixed Soluble length

Question

Let $s$ be a natural number. Is it possible to find, for each $n$ natural number greater than some arbitrary constant, a torsion-free group whose nilpotency class is $n$ and soluble length is $s$?

Answer 1

Take the semidirect product $G_n$ of ${\mathbb Z}^n$ generated by $x_1,\ldots,x_n$, with an infinite cyclic group $\langle y \rangle$, where $x_i^y=x_ix_{i+1}$ for $i<n$ and $x_n^y=x_n$. Then $G_n$ is nilpotent of class $n-1$ and solvable of length $2$.

For each $s>2$, let $H_s$ be a fixed torsion-free nilpotent group of solvable length $s$. Then $G_{n+1} \times H_s$ satisfies your conditions whenever $n$ is at least the nilpotency class of $H_s$.