Ingeometric group theory,Gromov's theorem on groups of polynomial growth, first proved byMikhail Gromov,^[1] characterizes finitely generatedgroups ofpolynomial growth, as those groups which havenilpotent subgroups of finiteindex.

Thegrowth rate of a group is awell-defined notion fromasymptotic analysis. To say that a finitely generated group haspolynomial growth means the number of elements oflength at mostn (relative to a symmetric generating set) is bounded above by apolynomial functionp(n). Theorder of growth is then the least degree of any such polynomial functionp.

Anilpotent groupG is a group with alower central series terminating in the identity subgroup.

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup that is of finite index.

There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result ofJoseph A. Wolf^[2] showed that ifG is a finitely generated nilpotent group, then the group has polynomial growth.Yves Guivarc'h^[3] and independentlyHyman Bass^[4] (with different proofs) computed the exact order of polynomial growth. LetG be a finitely generated nilpotent group with lower central series

${\displaystyle G=G_1\supseteq G_2\supseteq \cdots .}$ 

In particular, the quotient groupG_k/G_k+1 is a finitely generated abelian group.

TheBass–Guivarc'h formula states that the order of polynomial growth ofG is

${\displaystyle d(G)=\sum _{k\ge 1}k\mathrm{rank}(G_k/G_{k+1})}$ 

where:

rank denotes therank of an abelian group, i.e. the largest number of independent and torsion-free elements of the abelian group.

In particular, Gromov's theorem and the Bass–Guivarc'h formula imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity (excluding for example, fractional powers).

Another nice application of Gromov's theorem and the Bass–Guivarch formula is to thequasi-isometric rigidity of finitely generated abelian groups: any group which isquasi-isometric to a finitely generated abelian group contains a free abelian group of finite index.

In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now called theGromov–Hausdorff convergence, is currently widely used in geometry.

A relatively simple proof of the theorem was found byBruce Kleiner.^[5] Later,Terence Tao andYehuda Shalom modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds.^[6][7] Gromov's theorem also follows from the classification ofapproximate groups obtained by Breuillard, Green and Tao. A simple and concise proof based onfunctional analytic methods is given byOzawa.^[8]

Beyond Gromov's theorem one can ask whether there exists a gap in the growth spectrum for finitely generated group just above polynomial growth, separating virtually nilpotent groups from others. Formally, this means that there would exist a function${\displaystyle f:\mathbb{N}\to \mathbb{N}}$  such that a finitely generated group is virtually nilpotent if and only if its growth function is an${\displaystyle O(f(n))}$ . Such a theorem was obtained by Shalom and Tao, with an explicit function${\displaystyle n^{\log \log (n{)}^c}}$  for some${\displaystyle c>0}$ . All known groups with intermediate growth (i.e. both superpolynomial and subexponential) are essentially generalizations ofGrigorchuk's group, and have faster growth functions; so all known groups have growth faster than${\displaystyle e^{n^{\alpha }}}$ , with${\displaystyle \alpha =\log (2)/\log (2/\eta )\approx 0.767}$ , where${\displaystyle \eta }$  is the real root of the polynomial${\displaystyle x^3+x^2+x-2}$ .^[9]

It is conjectured that the true lower bound on growth rates of groups with intermediate growth is${\displaystyle e^{\sqrt{n}}}$ . This is known as theGap conjecture.^[10]

• Breuillard–Green–Tao theorem