A finitary version of Gromov's polynomial growth theorem

TL;DR

This paper proves a quantitative version of Gromov's polynomial growth theorem, showing that groups with polynomial growth at sufficiently large scales have virtually nilpotent subgroups, with explicit bounds provided.

Contribution

It introduces an explicit, finitary bound for Gromov's theorem, linking polynomial growth at large scales to virtual nilpotency with effective bounds.

Findings

Groups with polynomial growth at large scales have finite index nilpotent subgroups.

Provides explicit bounds on the index and step of the nilpotent subgroup.

Extends to finite groups with polycyclic conditions.

Abstract

We show that for some absolute (explicit) constant $C$, the following holds for every finitely generated group $G$, and all $d >0$: If there is some $ R_0 > \exp(\exp(Cd^C))$ for which the number of elements in a ball of radius $R_0$ in a Cayley graph of $G$ is bounded by $R_0^d$, then $G$ has a finite index subgroup which is nilpotent (of step $<C^d$). An effective bound on the finite index is provided if "nilpotent" is replaced by 'polycyclic", thus yielding a non-trivial result for finite groups as well.