# Gromov's theorem on groups of polynomial growth

In geometric group theory, **Gromov's theorem on groups of polynomial growth**, first proved by Mikhail Gromov,[1] characterizes finitely generated groups of *polynomial* growth, as those groups which have nilpotent subgroups of finite index.

## Statement

The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has **polynomial growth** means the number of elements of length (relative to a symmetric generating set) at most *n* is bounded above by a polynomial function *p*(*n*). The *order of growth* is then the least degree of any such polynomial function *p*.

A nilpotent group *G* is a group with a lower 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.

## Growth rates of nilpotent groups

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

In particular, the quotient group *G*_{k}/*G*_{k+1} is a finitely generated abelian group.

**The Bass–Guivarc'h formula** states that the order of polynomial growth of *G* is

where:

*rank*denotes the rank 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–Guivarch 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 the quasi-isometric rigidity of finitely generated abelian groups: any group which is quasi-isometric to a finitely generated abelian group contains a free abelian group of finite index.

## Proofs of Gromov's theorem

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

A relatively simple proof of the theorem was found by Bruce Kleiner.[5] Later, Terence Tao and Yehuda 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 of approximate groups obtained by Breuillard, Green and Tao.

## The gap conjecture

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
such that a finitely generated group is virtually nilpotent if and only if its growth function is an
. Such a theorem was obtained by Shalom and Tao, with an explicit function
for some
. The only known groups with growth functions both superpolynomial and subexponential (essentially generalisation of Grigorchuk's group) all have growth type of the form
, with
. Motivated by this it is natural to ask whether there are groups with growth type both superpolynomial and dominated by
. This is known as the *Gap conjecture*.[8]

## References

- Gromov, Mikhail (1981). With an appendix by Jacques Tits. "Groups of polynomial growth and expanding maps".
*Inst. Hautes Études Sci. Publ. Math*.**53**: 53–73. MR 0623534. - Wolf, Joseph A. (1968). "Growth of finitely generated solvable groups and curvature of Riemannian manifolds".
*Journal of Differential Geometry*.**2**(4): 421–446. MR 0248688. - Guivarc'h, Yves (1973). "Croissance polynomiale et périodes des fonctions harmoniques".
*Bull. Soc. Math. France*(in French).**101**: 333–379. MR 0369608. - Bass, Hyman (1972). "The degree of polynomial growth of finitely generated nilpotent groups".
*Proceedings of the London Mathematical Society*. Series 3.**25**(4): 603–614. doi:10.1112/plms/s3-25.4.603. MR 0379672. - Kleiner, Bruce (2010). "A new proof of Gromov's theorem on groups of polynomial growth".
*Journal of the American Mathematical Society*.**23**(3): 815–829. arXiv:0710.4593. Bibcode:2010JAMS...23..815K. doi:10.1090/S0894-0347-09-00658-4. MR 2629989. - Tao, Terence (2010-02-18). "A proof of Gromov's theorem".
*What’s new*. - Shalom, Yehuda; Tao, Terence (2010). "A finitary version of Gromov's polynomial growth theorem".
*Geom. Funct. Anal.***20**(6): 1502–1547. arXiv:0910.4148. doi:10.1007/s00039-010-0096-1. MR 2739001. - Grigorchuk, Rostislav I. (1991). "On growth in group theory".
*Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990)*. Math. Soc. Japan. pp. 325–338.