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| In [[geometric group theory]], '''Gromov's theorem on groups of polynomial growth''', named for [[Mikhail Gromov (mathematician)|Mikhail Gromov]], characterizes finitely generated [[Group (mathematics)|groups]] of ''polynomial'' growth, as those groups which have [[nilpotent group|nilpotent]] subgroups of finite [[index of a subgroup|index]].
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| The [[Growth rate (group theory)|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''.
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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 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 series
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| :<math> G = G_1 \supseteq G_2 \supseteq \ldots. </math> | |
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| In particular, the quotient group ''G''<sub>''k''</sub>/''G''<sub>''k''+1</sub> is a finitely generated abelian group.
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| '''The Bass–Guivarc'h formula''' states that the order of polynomial growth of ''G'' is
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| :<math> d(G) = \sum_{k \geq 1} k \ \operatorname{rank}(G_k/G_{k+1}) </math>
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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 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).
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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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| 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.<ref>http://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/</ref><ref>{{cite arxiv |eprint=0910.4148 |author1=Yehuda Shalom |author2=Terence Tao |title=A finitary version of Gromov's polynomial growth theorem |class=math.GR |year=2009}}</ref>
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| == References ==
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| <references/> | |
| * H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, ''Proceedings London Mathematical Society'', vol 25(4), 1972
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| * M. Gromov, Groups of Polynomial growth and Expanding Maps, [http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1981__53_ ''Publications mathematiques I.H.É.S.'', 53, 1981]
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| * Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A–B 272 (1971). [http://www.numdam.org/item?id=BSMF_1973__101__333_0]
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| * {{Cite arxiv | last1=Kleiner | first1=Bruce | year=2007 | title=A new proof of Gromov's theorem on groups of polynomial growth | arxiv=0710.4593}}
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| * J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, ''Journal of Differential Geometry'', vol 2, 1968
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| [[Category:Theorems in group theory]]
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| [[Category:Nilpotent groups]]
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| [[Category:Infinite group theory]]
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| [[Category:Metric geometry]]
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| [[Category:Geometric group theory]]
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