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- Gromov's_theorem_on_groups_of_polynomial_growth abstract "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 seriesIn particular, the quotient group Gk/Gk+1 is a finitely generated abelian group. The Bass–Guivarc'h formula states that the order of polynomial growth of G iswhere: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.".
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- Gromov's_theorem_on_groups_of_polynomial_growth hasPhotoCollection Gromov's_theorem_on_groups_of_polynomial_growth.
- Gromov's_theorem_on_groups_of_polynomial_growth subject Category:Geometric_group_theory.
- Gromov's_theorem_on_groups_of_polynomial_growth subject Category:Infinite_group_theory.
- Gromov's_theorem_on_groups_of_polynomial_growth subject Category:Metric_geometry.
- Gromov's_theorem_on_groups_of_polynomial_growth subject Category:Nilpotent_groups.
- Gromov's_theorem_on_groups_of_polynomial_growth subject Category:Theorems_in_group_theory.
- Gromov's_theorem_on_groups_of_polynomial_growth type Abstraction100002137.
- Gromov's_theorem_on_groups_of_polynomial_growth type Communication100033020.
- Gromov's_theorem_on_groups_of_polynomial_growth type Group100031264.
- Gromov's_theorem_on_groups_of_polynomial_growth type Message106598915.
- Gromov's_theorem_on_groups_of_polynomial_growth type NilpotentGroups.
- Gromov's_theorem_on_groups_of_polynomial_growth type Proposition106750804.
- Gromov's_theorem_on_groups_of_polynomial_growth type Statement106722453.
- Gromov's_theorem_on_groups_of_polynomial_growth type Theorem106752293.
- Gromov's_theorem_on_groups_of_polynomial_growth type TheoremsInAlgebra.
- Gromov's_theorem_on_groups_of_polynomial_growth type TheoremsInGroupTheory.
- Gromov's_theorem_on_groups_of_polynomial_growth comment "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.".
- Gromov's_theorem_on_groups_of_polynomial_growth label "Gromov's theorem on groups of polynomial growth".
- Gromov's_theorem_on_groups_of_polynomial_growth label "Théorème de Gromov sur les groupes à croissance polynomiale".
- Gromov's_theorem_on_groups_of_polynomial_growth label "Теорема Громова о группах полиномиального роста".
- Gromov's_theorem_on_groups_of_polynomial_growth sameAs Théorème_de_Gromov_sur_les_groupes_à_croissance_polynomiale.
- Gromov's_theorem_on_groups_of_polynomial_growth sameAs m.02z37w.
- Gromov's_theorem_on_groups_of_polynomial_growth sameAs Q3527091.
- Gromov's_theorem_on_groups_of_polynomial_growth sameAs Q3527091.
- Gromov's_theorem_on_groups_of_polynomial_growth sameAs Gromov's_theorem_on_groups_of_polynomial_growth.
- Gromov's_theorem_on_groups_of_polynomial_growth wasDerivedFrom Gromov's_theorem_on_groups_of_polynomial_growth?oldid=543588657.
- Gromov's_theorem_on_groups_of_polynomial_growth isPrimaryTopicOf Gromov's_theorem_on_groups_of_polynomial_growth.