Gromov's theorem on groups of polynomial growth

In mathematics, Gromov's theorem on groups of polynomial growth, named for Mikhail Gromov, characterizes finitely generated

Related Topics:
Mathematics - Mikhail Gromov

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groups of polynomial growth, as those groups which have nilpotent

Related Topics:
Groups - Nilpotent

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subgroups of finite index.

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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.

Related Topics:
Growth rate - Well-defined - Asymptotic analysis - Length - Polynomial - Degree

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A nilpotent group G is a group with a lower central series terminating in the identity subgroup.

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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.

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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

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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

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: G = G_1 supseteq G_2 supseteq ldots

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

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Bass's theorem states that the order of polynomial growth of G is

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

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where:

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:rank denotes the rank of an abelian group, i.e. the largest number of independent and torsion-free elements of the abelian group.

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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).

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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.

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