en>MarioS: +remark about analysis (no need to hide this in the main part)

2012-07-30T17:48:25Z

+remark about analysis (no need to hide this in the main part)

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In the [[Mathematics|mathematical]] area of [[group theory]], the '''Grigorchuk group''' or '''the first Grigorchuk group''' is a [[finitely generated group]] constructed by [[Rostislav Grigorchuk]] that provided the first example of a [[finitely generated group]] of intermediate (that is, faster than polynomial but slower than exponential) [[growth rate (group theory)|growth]]. The group was originally constructed by Grigorchuk in a 1980 paper<ref name="G0"/> and he then proved in a 1984 paper<ref name="G"/> that this group has intermediate growth, thus providing an answer to an important open problem posed by [[John Milnor]] in 1968. The Grigorchuk group remains a key object of study in [[geometric group theory]], particularly in the study of the so-called branch groups and automata groups, and it has important connections with the theory of [[iterated monodromy group]]s.<ref name="N">Volodymyr Nekrashevych. [http://books.google.com/books?hl=en&id=QG9f1Dn_RlwC&dq=Volodymyr+Nekrashevych.+%22Self-similar+groups%22&printsec=frontcover&source=web&ots=vzt5vnRb2U&sig=NSg79OI7JiQmTDqO-x7An5Nh5LM&sa=X&oi=book_result&resnum=1&ct=result ''Self-similar groups.''] Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-3831-8.</ref>

==History and generalizations==

The [[growth rate (group theory)|growth]] of a [[finitely generated group]] measures the asymptotics, as ''n'' → <math>\infty</math> of the size of an ''n''-ball in the [[Cayley graph]] of the group (that is, the number of elements of ''G'' that can be expressed as words of length at most ''n'' in the generating set of ''G''). The study of growth rates of [[finitely generated group]]s goes back to 1950s and is motivated in part by the notion of [[volume entropy]] (that is, the growth rate of the volume of balls) in the [[universal covering space]] of a [[compact space|compact]] [[Riemannian manifold]] in [[differential geometry]]. It is obvious that the growth rate of a finitely generated group is at most [[exponential function|exponential]] and it was also understood early on that finitely generated [[nilpotent group]]s have polynomial growth. In 1968 [[John Milnor]] posed a question<ref>John Milnor, Problem No. 5603, [[American Mathematical Monthly]], vol. 75 (1968), pp. 685–686.</ref> about the existence of a finitely generated group of ''intermediate growth'', that is, faster than any polynomial function and slower than any exponential function. An important result in the subject is [[Gromov's theorem on groups of polynomial growth]], obtained by [[Mikhail Gromov (mathematician)|Gromov]] in 1981, which shows that a finitely generated group has polynomial growth if and only if this group has a [[nilpotent group|nilpotent]] [[subgroup]] of finite [[index of a subgroup|index]]. Prior to Grigorchuk's work, there were many results establishing growth dichotomy (that is, that the growth is always either polynomial or exponential) for various classes of finitely generated groups, such as [[linear group]]s, [[solvable group]]s,<ref>[[John Milnor]]. [http://intlpress.com/JDG/archive/pdf/1968/2-4-447.pdf ''Growth of finitely generated solvable groups.''] [[Journal of Differential Geometry]]. vol. 2 (1968), pp. 447–449.</ref><ref>Joseph Rosenblatt. ''Invariant Measures and Growth Conditions'', [[Transactions of the American Mathematical Society]], vol. 193 (1974), pp. 33–53.</ref> etc.

Grigorchuk's group ''G'' was constructed in a 1980 paper of [[Rostislav Grigorchuk]],<ref name="G0">R. I. Grigorchuk. ''On Burnside's problem on periodic groups.'' (Russian) Funktsionalyi Analiz i ego Prilozheniya, vol. 14 (1980), no. 1, pp. 53–54.</ref> where he proved that this group is infinite, [[periodic group|periodic]] and [[residually finite group|residually finite]]. In a subsequent 1984 paper<ref name="G">R. I. Grigorchuk, ''Degrees of growth of finitely generated groups and the theory of invariant means.'' Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya. vol. 48 (1984), no. 5, pp. 939–985.</ref> Grigorchuk proved that this group has intermediate growth (this result was announced by Grigorchuk in 1983).<ref>{{cite journal| last=Grigorchuk | first=R. I. | year=1983 | title=К проблеме Милнора о групповом росте | trans_title=On the Milnor problem of group growth | language=Russian | journal=[[Doklady Akademii Nauk SSSR]] | volume=271 | issue=1 | pages=30–33}}</ref> More precisely, he proved that ''G'' has growth ''b''(''n'') that is faster than <math>\exp(\sqrt n)</math> but slower than exp(''n''''s'') where <math>s=\log_{32}31\approx 0.991</math>. The upper bound was later improved by [[Laurent Bartholdi]]<ref>Laurent Bartholdi. ''Lower bounds on the growth of a group acting on the binary rooted tree.'' International Journal of Algebra and Computation, vol. 11 (2001), no. 1, pp. 73–88.</ref> to <math>s=\log2/(\log2-\log\eta)\approx0.7675</math>, with <math>\eta^3+\eta^2+\eta=2</math>. A lower bound of <math>\exp(n^{0.504})</math> was proved by [[Yurii Leonov]].<ref>Yu. G. Leonov,
''On a lower bound for the growth of a 3-generator 2-group.'' Matematicheskii Sbornik, vol. 192 (2001), no. 11, pp. 77–92; translation in: Sbornik Mathematics. vol. 192 (2001), no. 11–12, pp. 1661–1676.</ref>

Grigorchuk's group was also the first example of a group that is [[amenable group|amenable]] but not [[elementary amenable group|elementary amenable]], thus answering a problem posed by [[Mahlon Day]] in 1957.<ref>Mahlon M. Day. ''Amenable semigroups.'' Illinois Journal of Mathematics, vol. 1 (1957), pp. 509–544.</ref>

Originally, Grigorchuk's group ''G'' was constructed as a group of Lebesgue-measure-preserving transformations on the unit interval, but subsequently simpler descriptions of ''G'' were found and it is now usually presented as a group of automorphisms of the infinite regular [[binary tree|binary]] [[rooted tree]]. The study of Grigorchuk's group informed in large part the development of the theory of branch groups, automata groups and self-similar groups in the 1990s–2000s and Grigorchuk's group remains a central object in this theory. Recently important connections between this theory and complex dynamics, particularly the notion of [[iterated monodromy group]]s, have been uncovered in the work of [[Volodymyr Nekrashevych]].<ref>Volodymyr Nekrashevych, [http://books.google.com/books?hl=en&id=QG9f1Dn_RlwC&dq=Volodymyr+Nekrashevych.+%22Self-similar+groups%22&printsec=frontcover&source=web&ots=vzt5vnRb2U&sig=NSg79OI7JiQmTDqO-x7An5Nh5LM&sa=X&oi=book_result&resnum=1&ct=result ''Self-similar groups.''] Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-3831-8.</ref> and others.

After Grigorchuk's 1984 paper, there were many subsequent extensions and generalizations,<ref>Roman Muchnik, and [[Igor Pak]]. ''On growth of Grigorchuk groups.''
International Journal of Algebra and Computation, vol. 11 (2001), no. 1, pp. 1–17.</ref><ref>Laurent Bartholdi. [http://imrn.oxfordjournals.org/cgi/reprint/1998/20/1049 ''The growth of Grigorchuk's torsion group.''] International Mathematics Research Notices, 1998, no. 20, pp. 1049–1054.</ref><ref>Anna Erschler.
[http://aif.cedram.org/fitem?id=AIF_2005__55_2_493_0 ''Critical constants for recurrence of random walks on G-spaces.''] Université de Grenoble. Annales de l'Institut Fourier, vol. 55 (2005), no. 2, pp. 493–509.</ref><ref>Jeremie Brieussel, [http://www.institut.math.jussieu.fr/theses/2008/brieussel/ Growth of certain groups], Doctoral Dissertation, University of Paris, 2008.</ref> though no improvement on the upper and lower bounds of the growth of the Grigorchuk group; the precise asymptotics of its growth is still unknown. It is conjectured that <math>\lim_{n\to \infty} \log_n \log b(n)</math> on the word growth exist, but even this remains a major open problem.

==Definition==
Although initially the Grigorchuk group was defined as a group of [[Lebesgue measure]]-preserving transformations of the unit interval, at present this group is usually given by its realization as a group of automorphisms of the infinite regular [[binary tree|binary]] [[rooted tree]] ''T''2.
The tree ''T''2 is realized as the set ''Σ*'' of all (including the empty string) finite strings in the alphabet ''Σ'' = {0,1}. The empty string Ø is the root vertex of ''T''2 and for a vertex ''x'' of ''T''2 the string ''x''0 is the [[Binary tree|left child]] of ''x'' and the string ''x''1 is the [[Binary tree|right child]] of ''x'' in ''T''2. The group of all automorphisms Aut(''T''2) can thus be thought of as the group of all length-preserving [[permutation group|permutations]] ''σ'' of ''Σ*'' that also respect the ''initial segment'' relation, that is such that whenever a string ''x'' is an initial segment of a string ''y'' then ''σ''(''x'') is an initial segment of ''σ''(''y'').

The '''Grigorchuk group''' ''G'' is then defined as the [[subgroup]] of Aut(''T''2) [[Generating set of a group|generated]] by four specific elements ''a,b,c,d'' of Aut(''T''2), that is ''G'' = <''a'',''b'',''c'',''d''> ≤ Aut(''T''2), where the automorphisms ''a,b,c,d'' of ''T''2 are defined recursively as follows:
*''a''(0''x'') = 1''x'', ''a''(1''x'') = 0''x'' for every ''x'' in ''Σ*'';
*''b''(0''x'') = 0''a''(''x''), ''b''(1''x'') = 1''c''(''x'') for every ''x'' in ''Σ*'';
*''c''(0''x'') = 0''a''(''x''), ''c''(1''x'') = 1''d''(''x'') for every ''x'' in ''Σ*'';
*''d''(0''x'') = 0''x'', ''d''(1''x'') = 1''b''(''x'') for every ''x'' in ''Σ*''.

Thus ''a'' swaps the right and left branch trees ''TL'' = 0Σ* and ''TR'' = 1Σ* below the root vertex ''Ø'' and the elements b,c,d can be represented as:
*''b'' = (''a'',''c''),
*''c'' = (''a'',''d''),
*''d'' = (1,''b'').
Here ''b'' = (''a'',''c'') means that ''b'' fixes the first level of ''T''2 (that is, it fixes the strings 0 and 1) and that ''b'' acts on ''TL'' exactly as the automorphism ''a'' does on ''T''2 and that ''b'' acts on ''TR'' exactly as the automorphism ''c'' does on ''T''2. The notation ''c'' = (''a'',''d'') and ''d'' = (1,''b'') is interpreted similarly, where 1 in ''d'' = (1,''b'') means that ''d'' acts on ''TL'' as the [[identity function|identity map]] does on ''T2''.

Of the four elements ''a, b, c, d'' of Aut(''T''2) only the element ''a'' is defined explicitly and the elements ''b, c, d'' are defined inductively (by induction on the length |''x''| of a string ''x'' in ''Σ*'' ), that is, level by level.

===Basic features of the Grigorchuk group===
The following are basic algebraic properties of the Grigorchuk group (see<ref name="DH">Pierre de la Harpe. [http://books.google.com/books?id=60fTzwfqeQIC&pg=PP1&dq=de+la+Harpe.+Topics+in+geometric+group+theory ''Topics in geometric group theory.''] Chicago Lectures in Mathematics. University of Chicago Press, Chicago. ISBN 0-226-31719-6; Ch. VIII, The first Grigorchuk group, pp. 211–264.</ref> for proofs):
*The group ''G'' is [[residually finite group|residually finite]]. Indeed, for every positive integer ''n'' let ''T''[''n''] be the finite subtree of ''T2'' which is the union of the first ''n'' levels of and let ''ρn:G→Aut(T''[''n'']'')'' be the restriction homomorphism that sends every element of G to its restriction to the finite tree ''T''[''n'']. The groups ''Aut(T''[''n'']'')'' are finite and for every nontrivial ''g'' in ''G'' there exists ''n'' such that ''ρn''(''g'') <math>\ne</math> 1.
*Each of the elements ''a,b,c,d'' has [[Order (group theory)|order]] 2 in ''G'', that is, ''a''2 = ''b''2 = ''c''2 = ''d''2 = 1. Thus ''a'' = ''a''−1, ''b'' = ''b''−1, ''c'' = ''c''−1 and ''d'' = ''d''−1, so that every element of ''G'' can be written as a positive word in ''a,b,c,d'', without using inverses.
*The elements ''b,c,d'' pairwise commute and ''bc = cb = d'', ''bd = db = c'', ''dc = dc = b'', so that <''b,c,d''>≤''G'' is an [[abelian group]] of order 4 [[group isomorphism|isomorphic]] to the [[direct product of groups|direct product]] of two [[cyclic group]]s of order 2.
*The group ''G'' is generated by ''a'' and any two of the three elements ''b,c,d'' (e.g. ''G'' = <''a'', ''b'', c>).
*Using the above recursive notation, in ''G'' we have ''aba'' = (''c'',''a''), ''aca'' = (''d'',''a''), ''ada'' = (''b'',1).
*The [[group action|stabilizer]] St''G''[1] in ''G'' of the 1st level of ''T''2 is the subgroup generated by ''b, c, d, aba, aca, ada''. The subgroup St''G''[1] is a [[normal subgroup]] of [[index of a subgroup|index]] 2 in ''G'' and
:''G'' = St''G''[1] <math>\sqcup</math> ''a'' St''G''[1].
*Every element of ''G'' can be written as a (positive) word in ''a,b,c,d'' such that this word does not contain any subwords of the form ''aa, bb, cc, dd, cd, dc, bc, cb, bd, db''. Such words are called ''reduced''.
*A reduced word represents an element of St''G''[1] if and only if this word involves an even number of occurrences of ''a''.
*If ''w'' is a reduced word of even length involving a positive even number of occurrences of ''a'' then there are some words ''u,v'' in ''a,b,c,d'' (not necessarily reduced) such that in ''G'' we have ''w'' = (''u'',''v'') and such that |''u''| ≤ |''w''|/2, |''v''| ≤ |''w''|/2. A similar statement holds if ''w'' is a reduced word of odd length involving a positive even number of occurrences of ''a'' where in the conclusion we have |''u''| ≤ (|''w''| + 1)/2, |''v''| ≤ (|''w''| + 1)/2.

The last property of ''G'' is sometimes called the ''contraction property''. This property plays a key role in many proofs regarding ''G'' since it allows to use inductive arguments on the length of a word.

==Properties and facts regarding the Grigorchuk group<ref name="DH"/>==
*The group ''G'' is infinite.<ref name="G"/>
*The group ''G'' is residually finite.<ref name="G"/>
*The group ''G'' is a [[p-group|2-group]], that is, every element in G has finite [[Order (group theory)|order]] that is a power of 2.<ref name="G0"/>
*The group ''G'' has intermediate growth.<ref name="G"/>
*The group ''G'' is [[amenable group|amenable]] but not [[elementary amenable group|elementary amenable]].<ref name="G"/>
*The group ''G'' is ''[[just infinite group|just infinite]]'', that is G is infinite but every proper [[quotient group]] of ''G'' is finite.
*The group ''G'' has the ''congruence subgroup property'': if ''H ≤ G'' then H has finite [[index of a subgroup|index]] in ''G'' if and only if there is a positive integer ''n'' such that the [[kernel of a homomorphism|kernel]] Kern(''ρ''''n'') of ''ρn'' is contained in ''H'', that is Kern(''ρn'') ≤ ''H''.
*The group ''G'' has solvable [[Word problem for groups|word problem]] and solvable [[conjugacy problem]] (following from the contraction property mentioned in the previous section).
*The group ''G'' has solvable [[subgroup membership problem]], that is, there is an algorithm that, given arbitrary words ''w'', ''u''1,..., ''un'' in ''a, b, c, d'', decides whether or not ''w'' represents an element of the subgroup generated by ''u''1,..., ''un'' in ''G''.<ref name="GW"/>
*The group ''G'' is [[LERF|subgroup separable]], that is, every finitely generated subgroup is closed in the [[Profinite group|pro-finite topology]] on ''G''.<ref name="GW">R. I.Grigorchuk, and J. S. Wilson. [http://journals.cambridge.org/action/displayAbstract;jsessionid=61D43BDA6A49AF3EA6074EE64ADB3071.tomcat1?fromPage=online&aid=186399 ''A structural property concerning abstract commensurability of subgroups.''] [[Journal of the London Mathematical Society]] (2), vol. 68 (2003), no. 3, pp. 671–682.</ref>
*Every [[maximal subgroup]] of ''G'' has finite [[index of a subgroup|index]] in ''G''.<ref>E. L. Pervova,
''Everywhere dense subgroups of a group of tree automorphisms.'' (in Russian). Trudy Matematicheskogo Instituta Imeni V. A. Steklova. vol. 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, pp. 356–367; translation in: Proceedings of the Steklov Institute of Mathematics, vol 231 (2000), no. 4, pp. 339–350.</ref>
*The group ''G'' is finitely generated but not [[finitely presented group|finitely presentable]].<ref name="G"/><ref>I. G. Lysënok, ''A set of defining relations for the Grigorchuk group.'' Matematicheskie Zametki, vol. 38 (1985), no. 4, pp. 503–516.</ref>

==See also==
*[[Geometric group theory]]
*[[Growth rate (group theory)|Growth of finitely generated groups]]
*[[Amenable group]]s
*[[Iterated monodromy group]]

==References==
{{reflist}}

==External links==
*[http://arxiv.org/abs/math.GR/0607384 Rostislav Grigorchuk and Igor Pak, ''Groups of Intermediate Growth: an Introduction for Beginners'', preprint, 2006, arXiv]

{{DEFAULTSORT:Grigorchuk Group}}
[[Category:Group theory]]
[[Category:Geometric group theory]]

GGH encryption scheme - Revision history

en>MarioS: +remark about analysis (no need to hide this in the main part)