 One-scale virtual nilpotence theorem

For every $d>0$ there is $N(d)$ such that the following holds. Suppose a group $G$ has a finite symmetric generating set $S$ containing the identity, andThen there are subgroupswith $H\subseteq S^{\lfloor n/2\rfloor }$, bounded index of a subgroup $[G:C]=O_d(1)$, and nilpotent quotient group $C/H$ of class $O_d(1)$. In particular $H$ is finite. The containment allows $C=G$.^[ref]

A geometric-scale pigeonhole argument finds an intermediate word metric ball with bounded growth. The Breuillard-Green-Tao structure theorem for approximate groups then supplies the finite subgroup and the bounded-index nilpotent quotient.

For example, take a cyclic group of arbitrarily large prime order and $S=G$. The growth hypothesis holds, but every proper subgroup has index equal to that prime. Requiring a proper $C$ would therefore contradict an index bound depending only on $d$; choosing $C=G$ gives the valid nilpotent conclusion.