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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Grigorchuk group ： ウィキペディア英語版 Grigorchuk group In the 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. The group was originally constructed by Grigorchuk in a 1980 paper〔 and he then proved in a 1984 paper〔 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 groups.〔Volodymyr Nekrashevych. (''Self-similar groups.'' ) Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-3831-8.〕 ==History and generalizations== The growth of a finitely generated group measures the asymptotics, as ''n'' → $\backslash infty$ 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 groups 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 Riemannian manifold in differential geometry. It is obvious that the growth rate of a finitely generated group is at most exponential and it was also understood early on that finitely generated nilpotent groups have polynomial growth. In 1968 John Milnor posed a question〔John Milnor, Problem No. 5603, American Mathematical Monthly, vol. 75 (1968), pp. 685–686.〕 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 Gromov in 1981, which shows that a finitely generated group has polynomial growth if and only if this group has a nilpotent subgroup of finite 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 groups, solvable groups,〔John Milnor. (''Growth of finitely generated solvable groups.'' ) Journal of Differential Geometry. vol. 2 (1968), pp. 447–449.〕〔Joseph Rosenblatt. ''Invariant Measures and Growth Conditions'', Transactions of the American Mathematical Society, vol. 193 (1974), pp. 33–53.〕 etc. Grigorchuk's group ''G'' was constructed in a 1980 paper of Rostislav Grigorchuk,〔R. I. Grigorchuk. ''On Burnside's problem on periodic groups.'' (Russian) Funktsionalyi Analiz i ego Prilozheniya, vol. 14 (1980), no. 1, pp. 53–54.〕 where he proved that this group is infinite, periodic and residually finite. In a subsequent 1984 paper〔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.〕 Grigorchuk proved that this group has intermediate growth (this result was announced by Grigorchuk in 1983). More precisely, he proved that ''G'' has growth ''b''(''n'') that is faster than $\backslash exp(\backslash sqrt\; n)$ but slower than exp(''n''''s'') where $s=\backslash log\_31\backslash approx\; 0.991$. The upper bound was later improved by Laurent Bartholdi〔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.〕 to $s=\backslash log2/(\backslash log2-\backslash log\backslash eta)\backslash approx0.7675$, with $\backslash eta^3+\backslash eta^2+\backslash eta=2$. A lower bound of $\backslash exp(n^)$ was proved by Yurii Leonov.〔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.〕 Grigorchuk's group was also the first example of a group that is amenable but not elementary amenable, thus answering a problem posed by Mahlon Day in 1957.〔Mahlon M. Day. ''Amenable semigroups.'' Illinois Journal of Mathematics, vol. 1 (1957), pp. 509–544.〕 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 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 groups, have been uncovered in the work of Volodymyr Nekrashevych.〔Volodymyr Nekrashevych, (''Self-similar groups.'' ) Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. ISBN 0-8218-3831-8.〕 and others. After Grigorchuk's 1984 paper, there were many subsequent extensions and generalizations,〔Roman Muchnik, and Igor Pak. ''On growth of Grigorchuk groups.'' International Journal of Algebra and Computation, vol. 11 (2001), no. 1, pp. 1–17.〕〔Laurent Bartholdi. (''The growth of Grigorchuk's torsion group.'' ) International Mathematics Research Notices, 1998, no. 20, pp. 1049–1054.〕〔Anna Erschler. (''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.〕〔Jeremie Brieussel, (Growth of certain groups ), Doctoral Dissertation, University of Paris, 2008.〕 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 $\backslash lim\_\; \backslash log\_n\; \backslash log\; b(n)$ on the word growth exist, but even this remains a major open problem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Grigorchuk group」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.