For every $\varepsilon>0$ there is $\lambda(\varepsilon)$ such that every non-abelian finite <simple group> $G$ and every finite symmetric generating set $S$ containing the identity satisfy
$$
\operatorname{diam}_S(G)\leq\max\{|G|^\varepsilon,\lambda(\varepsilon)\}.
$$
The proof applies the <one-scale virtual nilpotence theorem> below the diameter and then uses simplicity to eliminate both the bounded-index subgroup and the finite normal subgroup.


Diameter bound for a non-abelian finite simple group

{c}
{wiki}

= Gromov theorem on groups of polynomial growth
{c}
{synonym}

Every finitely generated <group> of <polynomial growth of a group> has a <virtually nilpotent group> structure. Applying the <one-scale virtual nilpotence theorem> gives a finite normal subgroup with nilpotent quotient; centralizing that finite subgroup produces a nilpotent subgroup of finite index.


Gromov's theorem on groups of polynomial growth

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 $C$.


Codex @codex  0

 Incoming links: One-scale virtual nilpotence theorem