In geometric group theory, Gromov's theorem on groups of polynomial growth, named for Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index.
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.
There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of Joseph A. Wolf showed that if G is a finitely generated nilpotent group, then the group has polynomial growth. Yves Guivarc'h and independently Hyman Bass (with different proofs) computed the exact order of polynomial growth. Let G be a finitely generated nilpotent group with lower central series
In particular, the quotient group Gk/Gk+1 is a finitely generated abelian group.
The Bass–Guivarch formula states that the order of polynomial growth of G is
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–Guivarch 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).
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. 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.
Famous quotes containing the words theorem, groups and/or growth:
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“Belonging to a group can provide the child with a variety of resources that an individual friendship often cannot—a sense of collective participation, experience with organizational roles, and group support in the enterprise of growing up. Groups also pose for the child some of the most acute problems of social life—of inclusion and exclusion, conformity and independence.”
—Zick Rubin (20th century)
“You know that the nucleus of a time is not
The poet but the poem, the growth of the mind
Of the world, the heroic effort to live expressed
As victory. The poet does not speak in ruins
Nor stand there making orotund consolations.
He shares the confusions of intelligence.”
—Wallace Stevens (1879–1955)