Gromov's theorem on groups of polynomial growth explained

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 at most n (relative to a symmetric generating set) 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 … .

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} 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

f: N \to 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 n ^{loglog (n) ^{c}} for some 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 e ^{n ^{\alpha}} , with \alpha = log ( 2 )/ log ( 2 / η ) ≈ 0 . 767 , where η is the real root of the polynomial x ^{3+x} ^{2+x-2} .^[9]

It is conjectured that the true lower bound on growth rates of groups with intermediate growth is

e ^{\sqrt} . This is known as the Gap conjecture.^[10]

See also

• Breuillard–Green–Tao theorem

References

1. Gromov . Mikhail . Groups of polynomial growth and expanding maps . Inst. Hautes Études Sci. Publ. Math. . 1981 . 53 . . 53–73 . 10.1007/BF02698687 . 623534. 121512559 .
2. Wolf . Joseph A. . Growth of finitely generated solvable groups and curvature of Riemannian manifolds . . 2 . 1968 . 421–446 . 0248688 . 4 . 10.4310/jdg/1214428658 . free .
3. Yves . Guivarc'h . Croissance polynomiale et périodes des fonctions harmoniques . Bull. Soc. Math. France . 101 . 1973 . fr . 333–379 . 10.24033/bsmf.1764 . 0369608. free .
4. Hyman . Bass . The degree of polynomial growth of finitely generated nilpotent groups . . Series 3 . 25 . 4 . 1972 . 603–614 . 0379672 . 10.1112/plms/s3-25.4.603.
5. Kleiner . Bruce . 2010 . A new proof of Gromov's theorem on groups of polynomial growth . 0710.4593 . . 23 . 3 . 815–829 . 2629989 . 10.1090/S0894-0347-09-00658-4. 2010JAMS...23..815K . 328337 .
6. Web site: A proof of Gromov's theorem . Tao . Terence . 2010-02-18 . What’s new.
7. 0910.4148 . Yehuda . Shalom . Terence . Tao . A finitary version of Gromov's polynomial growth theorem . 2010 . 2739001 . . 20 . 6 . 1502–1547 . 10.1007/s00039-010-0096-1. 115182677 .
8. Ozawa . Narutaka . 2018 . A functional analysis proof of Gromov's polynomial growth theorem . 1510.04223 . . 51 . 3 . 549–556 . 3831031 . 10.24033/asens.2360 . 119278398 .
9. 1802.09077. Erschler. Anna. Anna Erschler. Zheng. Tianyi. Tianyi Zheng. Growth of periodic Grigorchuk groups. 2018.
10. Book: Grigorchuk, Rostislav I. . On growth in group theory . Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) . 325 - 338 . Math. Soc. Japan . 1991.