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 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

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

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]

See also

• Breuillard–Green–Tao theorem