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. Mathematics is the study of quantity, structure, space and change. ... See Mikhail Gromov (disambiguation) for other people with this name. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

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. In group theory, the growth rate of a group with respect to a symmetric generating set is a notion that describes how fast a group grows. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... This article is about the term degree as used in mathematics. ...

A nilpotent group G is a group with a lower central series terminating in the identity subgroup. In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ...

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

In particular, the quotient group G_k/G_k+1 is a finitely generated abelian group.

Bass's theorem 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 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 mathematics, the rank, or torsion-free rank, of an abelian group measures how large a group is in terms of how large a vector space one would need to contain it; or alternatively how large a free abelian group it can contain as a subgroup. ...

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. Gromov-Hausdorff convergence is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence. ...

References

• H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proceedings London Mathematical Society, vol 25(3), 1972
• M. Gromov, Groups of Polynomial growth and Expanding Maps, Publications mathematiques I.H.É.S., 53, 1981 (http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1981__53_)