Gromov's theorem on groups of polynomial growth Guide, Meaning , Facts, Information and Description
In mathematics, 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 degree of the 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 Hyman Bass showed that if G is a finitely generated nilpotent group, then the group has polynomial growth. Let G be a finitely generated nilpotent group with lower central series
Bass's theorem states that the order of polynomial growth of G is
In particular, Gromov's and Bass's theorems 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.