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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 (mathematics)|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.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


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
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:<math> G = G_1 \supseteq G_2 \supseteq \ldots. </math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


In particular, the quotient group ''G''<sub>''k''</sub>/''G''<sub>''k''+1</sub> is a finitely generated abelian group.  
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


'''The Bass&ndash;Guivarch formula''' states that the order of polynomial growth of ''G'' is
==Demos==


:<math> d(G) = \sum_{k \geq 1} k \ \operatorname{rank}(G_k/G_{k+1}) </math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


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 theorem and the Bass&ndash;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).
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now called  the [[Gromov&ndash;Hausdorff convergence]], is currently widely used in geometry.
==Test pages ==


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>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


== References ==
*[[Inputtypes|Inputtypes (private Wikis only)]]
<references/>
*[[Url2Image|Url2Image (private Wikis only)]]
* H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, ''Proceedings London Mathematical Society'', vol 25(4), 1972
==Bug reporting==
* 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]
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
* Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A&ndash;B 272 (1971). [http://www.numdam.org/item?id=BSMF_1973__101__333_0]
* {{Cite arxiv | last1=Kleiner | first1=Bruce | year=2007  | title=A new proof of Gromov's theorem on groups of polynomial growth | arxiv=0710.4593}}
* J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, ''Journal of Differential Geometry'', vol 2, 1968
 
[[Category:Theorems in group theory]]
[[Category:Nilpotent groups]]
[[Category:Infinite group theory]]
[[Category:Metric geometry]]
[[Category:Geometric group theory]]
 
[[fr:Théorème de Gromov sur les groupes à croissance polynomiale]]
[[ru:Теорема Громова о группах полиномиального роста]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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