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.

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.

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

$G = G_1 \supseteq G_2 \supseteq \ldots.$

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

$d(G) = \sum_{k \geq 1} k \ \operatorname{rank}(G_k/G_{k+1})$

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).

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]

References

1. 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 623534.
2. Wolf, Joseph A. (1968). "Growth of finitely generated solvable groups and curvature of Riemanniann manifolds". Journal of Differential Geometry 2 (4): 421–446. MR 0248688.
3. Guivarc'h, Yves (1973). "Croissance polynomiale et périodes des fonctions harmoniques". Bull. Soc. Math. France (in French) 101: 333–379. MR 0369608.
4. Bass, Hyman (1972). "The degree of polynomial growth of finitely generated nilpotent groups". Proceedings of the London Mathematical Society (3) 25 (4): 603–614. doi:10.1112/plms/s3-25.4.603. MR 0379672.
5. 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. doi:10.1090/S0894-0347-09-00658-4. MR 2629989.
6. Tao, Terence (2010-02-18). "A proof of Gromov’s theorem". What’s new.
7. 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.