Large lower-step product in a torsion-free nilpotent approximate group

If $A$ is a finite $K$-approximate group in a torsion-free $s$-step nilpotent group, then there are $r\le K^{O(1)}$ approximate groups $A_0,\dots ,A_r\subseteq A^{O(1)}$, each with approximation parameter $K^{O(1)}$ and each generating a group of class less than $s$, such thatThe proof applies the large-progression form of the Freiman-Green-Ruzsa theorem in the abelianization, lifts its subgroup and cyclic directions, and uses the intersection of an approximate group power with a subgroup.

Let $\pi :G\to G/[G,G]$ be the abelianization map and let $A$ be a finite $K$-approximate group. Suppose a coset progression $HP(x_1,\dots ,x_r;L_1,\dots ,L_r)$ lies in $\pi (A^4)$ and has size at least $\delta |\pi (A)|$. ThenA section of $\pi$ is multiplicative up to a bounded power of $A$ inside the commutator subgroup; successively separating the subgroup and progression coordinates proves the inclusion needed for this estimate.

Let $\pi :G\to G/N$ be a quotient group map, let $A$ be finite and symmetric, and suppose $P\subseteq \pi (A^m)$ has $|P|\ge \delta |\pi (A)|$. ThenChoose one lift in $A^m$ of each member of $P$ and multiply those lifts by $A^2\cap N$. Distinct fibers are disjoint, while $|\pi (A)||A^2\cap N|\ge |A|$.