Английская Википедия: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> 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> 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>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>O(f(n))</math>. Such a theorem was obtained by Shalom and Tao, with an explicit function <math>n^{\log\log(n)^c}</math> for some <math>c > 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>e^{n^\alpha}</math>, with <math>\alpha = \log(2)/\log(2/\eta ) \approx 0.767</math>, where <math>\eta</math> is the real root of the polynomial <math>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>e^{\sqrt n}</math>. This is known as the Gap conjecture.[10]
References
