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
- [math]\displaystyle{ G = G_1 \supseteq G_2 \supseteq \cdots. }[/math]
In particular, the quotient group Gk/Gk+1 is a finitely generated abelian group.
The Bass–Guivarc'h formula states that the order of polynomial growth of G is
- [math]\displaystyle{ d(G) = \sum_{k \geq 1} k \operatorname{rank}(G_k/G_{k+1}) }[/math]
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 [math]\displaystyle{ f: \mathbb N \to \mathbb N }[/math] such that a finitely generated group is virtually nilpotent if and only if its growth function is an [math]\displaystyle{ O(f(n)) }[/math]. Such a theorem was obtained by Shalom and Tao, with an explicit function [math]\displaystyle{ n^{\log\log(n)^c} }[/math] for some [math]\displaystyle{ c \gt 0 }[/math]. 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 [math]\displaystyle{ e^{n^\alpha} }[/math], with [math]\displaystyle{ \alpha = \log(2)/\log(2/\eta ) \approx 0.767 }[/math], where [math]\displaystyle{ \eta }[/math] is the real root of the polynomial [math]\displaystyle{ x^3+x^2+x-2 }[/math].[9]
It is conjectured that the true lower bound on growth rates of groups with intermediate growth is [math]\displaystyle{ e^{\sqrt n} }[/math]. This is known as the Gap conjecture.[10]
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. doi:10.1007/BF02698687. http://www.numdam.org/item?id=PMIHES_1981__53__53_0.
- ↑ 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. http://projecteuclid.org/euclid.jdg/1214428658.
- ↑ Guivarc'h, Yves (1973). "Croissance polynomiale et périodes des fonctions harmoniques" (in fr). Bull. Soc. Math. France 101: 333–379. doi:10.24033/bsmf.1764. http://www.numdam.org/item?id=BSMF_1973__101__333_0.
- ↑ 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.
- ↑ 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. doi:10.1090/S0894-0347-09-00658-4. Bibcode: 2010JAMS...23..815K.
- ↑ Tao, Terence (2010-02-18). "A proof of Gromov's theorem". https://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/.
- ↑ Shalom, Yehuda; Tao, Terence (2010). "A finitary version of Gromov's polynomial growth theorem". Geom. Funct. Anal. 20 (6): 1502–1547. doi:10.1007/s00039-010-0096-1.
- ↑ Ozawa, Narutaka (2018). "A functional analysis proof of Gromov's polynomial growth theorem". Annales Scientifiques de l'École Normale Supérieure 51 (3): 549–556. doi:10.24033/asens.2360.
- ↑ Erschler, Anna; Zheng, Tianyi (2018). "Growth of periodic Grigorchuk groups". arXiv:1802.09077.
- ↑ 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.
 |  |
