Gromov's theorem on groups of polynomial growth

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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 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. Hyman Bass computed the exact order of polynomial growth. Let G be a finitely generated nilpotent group with lower central series

$G = G_1 \supseteq G_2 \supseteq \ldots$

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

$d(G) = \sum_{k \geq 1} k \ \operatorname{rank}(G_k/G_{k+1})$

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 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.

[edit] References

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proceedings London Mathematical Society, vol 25(4), 1972
M. Gromov, Groups of Polynomial growth and Expanding Maps, Publications mathematiques I.H.É.S., 53, 1981
J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, Journal of Differential Geometry, vol 2, 1968