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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 Growth rate (group theory) ： ウィキペディア英語版 Growth rate (group theory) In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length ''n''. ==Definition== Suppose ''G'' is a finitely generated group; and ''T'' is a finite ''symmetric'' set of generators (symmetric means that if $x\; \backslash in\; T$ then $x^\; \backslash in\; T$). Any element $x\; \backslash in\; G$ can be expressed as a word in the ''T''-alphabet :$x\; =\; a\_1\; \backslash cdot\; a\_2\; \backslash cdots\; a\_k\; \backslash mbox\; a\_i\backslash in\; T.$ Let us consider the subset of all elements of ''G'' which can be presented by such a word of length ≤ ''n'' :$B\_n(G,T)\; =\; \backslash \; k\backslash le\; n\backslash \}.$ This set is just the closed ball of radius ''n'' in the word metric ''d'' on ''G'' with respect to the generating set ''T'': :$B\_n(G,T)\; =\; \backslash .$ More geometrically, $B\_n(G,T)$ is the set of vertices in the Cayley graph with respect to ''T'' which are within distance ''n'' of the identity. Given two nondecreasing positive functions ''a'' and ''b'' one can say that they are equivalent ($a\backslash sim\; b$) if there is a constant ''C'' such that :$a(n/\; C)\; \backslash leq\; b(n)\; \backslash leq\; a(Cn),\backslash ,$ for example $p^n\backslash sim\; q^n$ if $p,q>1$. Then the growth rate of the group ''G'' can be defined as the corresponding equivalence class of the function :$\backslash \#(n)=|B\_n(G,T)|,$ where $|B\_n(G,T)|$ denotes the number of elements in the set $B\_n(G,T)$. Although the function $\backslash \#(n)$ depends on the set of generators ''T'' its rate of growth does not (see below) and therefore the rate of growth gives an invariant of a group. The word metric ''d'' and therefore sets $B\_n(G,T)$ depend on the generating set ''T''. However, any two such metrics are ''bilipschitz'' ''equivalent'' in the following sense: for finite symmetric generating sets ''E'', ''F'', there is a positive constant ''C'' such that :$\backslash \; d\_F(x,y)\; \backslash leq\; d\_(x,y)\; \backslash leq\; C\; \backslash \; d\_F(x,y).$ As an immediate corollary of this inequality we get that the growth rate does not depend on the choice of generating set. ==Polynomial and exponential growth== If :$\backslash \#(n)\backslash le\; C(n^k+1)$ for some we say that ''G'' has a polynomial growth rate. The infimum $k\_0$ of such ''ks is called the order of polynomial growth. According to Gromov's theorem, a group of polynomial growth is virtually nilpotent, i.e. it has a nilpotent subgroup of finite index. In particular, the order of polynomial growth $k\_0$ has to be a natural number and in fact $\backslash \#(n)\backslash sim\; n^$. If $\backslash \#(n)\backslash ge\; a^n$ for some $a>1$ we say that ''G'' has an exponential growth rate. Every finitely generated ''G'' has at most exponential growth, i.e. for some $b>1$ we have $\backslash \#(n)\backslash le\; b^n$. If $\backslash \#(n)$ grows more slowly than any exponential function, ''G'' has a subexponential growth rate. Any such group is amenable. 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「Growth rate (group theory)」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.