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

${\displaystyle G=G_{1}\supseteq G_{2}\supseteq \cdots .}$ 

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

${\displaystyle d(G)=\sum _{k\geq 1}k\operatorname {rank} (G_{k}/G_{k+1})}$ 

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–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 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. A simple and concise proof based on functional analytic methods is given by Ozawa.^[8]

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 ${\displaystyle O(f(n))}$ . 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 of Grigorchuk'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 the Gap conjecture.^[10]

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. doi:10.1007/BF02698687. MR 0623534. S2CID 121512559.
2. Wolf, Joseph A. (1968). "Growth of finitely generated solvable groups and curvature of Riemannian manifolds". Journal of Differential Geometry. 2 (4): 421–446. doi:10.4310/jdg/1214428658. MR 0248688.
3. Guivarc'h, Yves (1973). "Croissance polynomiale et périodes des fonctions harmoniques". Bull. Soc. Math. France (in French). 101: 333–379. doi:10.24033/bsmf.1764. MR 0369608.
4. 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.
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. Bibcode:2010JAMS...23..815K. doi:10.1090/S0894-0347-09-00658-4. MR 2629989. S2CID 328337.
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. S2CID 115182677.
8. Ozawa, Narutaka (2018). "A functional analysis proof of Gromov's polynomial growth theorem". Annales Scientifiques de l'École Normale Supérieure. 51 (3): 549–556. arXiv:1510.04223. doi:10.24033/asens.2360. MR 3831031. S2CID 119278398.
9. Erschler, Anna; Zheng, Tianyi (2018). "Growth of periodic Grigorchuk groups". arXiv:1802.09077.
10. 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.