= One-scale virtual nilpotence theorem
For every $d>0$ there is $N(d)$ such that, if a group has a finite symmetric generating set $S$ containing the identity and
$$
|S^n|\leq n^d|S|
$$
for some $n\geq N(d)$, then there are $H\trianglelefteq C<G$ with $H\subseteq S^{\lfloor n/2\rfloor}$, $[G:C]=O_d(1)$, and $C/H$ nilpotent of class $O_d(1)$. A geometric-scale pigeonhole argument finds a large radius of small tripling, and the <Breuillard-Green-Tao structure theorem for approximate groups> supplies $H$ and $C$.
Back to article page
