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

$G=G_{1}\supseteq G_{2}\supseteq \ldots .$

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).

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 ${\displaystyle f:\mathbb {N} \to \mathbb {N} }$ such that a finitely generated group is virtually nilpotent if and only if its growth function is an $O(f(n))$. Such a theorem was obtained by Shalom and Tao, with an explicit function ${\displaystyle n^{\log \log(n)^{c}}}$ for some $c>0$. The only known groups with growth functions both superpolynomial and subexponential (essentially generalisation of Grigorchuk's group) all have growth type of the form ${\displaystyle e^{n^{c}}}$, with ${\displaystyle 1/2<c<1}$. Motivated by this it is natural to ask whether there are groups with growth type both superpolynomial and dominated by ${\displaystyle e^{\sqrt {n}}}$. This is known as the Gap conjecture.^[8]

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.
8. 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.