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| {{Refimprove|date=March 2011}}
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| In [[group theory]], the '''growth rate''' of a [[group (mathematics)|group]] with respect to a symmetric [[generating set of a group|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''.
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| ==Definition==
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| Suppose ''G'' is a finitely generated group; and ''T'' is a finite ''symmetric'' set of [[Generating set of a group|generator]]s
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| (symmetric means that if <math> x \in T </math> then <math> x^{-1} \in T </math>).
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| Any element <math> x \in G </math> can be expressed as a [[string (computer science)#Formal theory|word]] in the ''T''-alphabet
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| :<math> x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T. </math> | |
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| Let us consider the subset of all elements of ''G'' which can be presented by such a word of length ≤ ''n''
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| :<math>B_n(G,T) = \{x\in G | x = a_1 \cdot a_2 \cdots a_k \mbox{ where } a_i\in T \mbox{ and } k\le n\}.</math>
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| This set is just the [[Ball (mathematics)|closed ball]] of radius ''n'' in the [[word metric]] ''d'' on ''G'' with respect to the generating set ''T'':
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| :<math>B_n(G,T) = \{x\in G | d(x, e)\le n\}.</math>
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| More geometrically, <math>B_n(G,T)</math> is the set of vertices in the [[Cayley graph]] with respect to ''T'' which are within distance ''n'' of the identity.
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| Given two nondecreasing positive functions ''a'' and ''b'' one can say that
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| they are equivalent (<math>a\sim b</math>) if there is a constant ''C'' such that
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| :<math> a(n/ C) \leq b(n) \leq a(Cn),\, </math>
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| for example <math> p^n\sim q^n </math> if <math> p,q>1 </math>.
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| Then the growth rate of the group ''G'' can be defined as the corresponding [[equivalence class]] of the function
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| :<math>\#(n)=|B_n(G,T)|, </math>
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| where <math>|B_n(G,T)|</math> denotes the number of elements in the set <math>B_n(G,T)</math>.
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| Although the function <math>\#(n)</math> depends on the set of generators ''T'' its rate of
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| growth does not (see below) and therefore the rate of growth gives an invariant of a group.
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| The word metric ''d'' and therefore sets <math>B_n(G,T)</math> depend
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| on the generating set ''T''. However, any two such metrics are [[Lipschitz continuity#Lipschitz continuity in metric spaces|''bilipschitz'']] [[equivalence class|''equivalent'']] in the following sense: for finite symmetric generating sets ''E'', ''F'', there is a positive constant ''C'' such that
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| :<math> {1\over C} \ d_F(x,y) \leq d_{E}(x,y) \leq C \ d_F(x,y). </math>
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| As an immediate corollary of this inequality we get that the growth rate does not depend on the choice of generating set. | |
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| ==Polynomial and exponential growth==<!-- This section is linked from [[Hyperbolic geometry]] -->
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| If
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| :<math>\#(n)\le C(n^k+1)</math>
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| for some <math>C,k<\infty</math> we say that ''G'' has a '''polynomial growth rate'''.
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| The infimum <math>k_0</math> of such ''k'''s is called the '''order of polynomial growth'''.
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| According to [[Gromov's theorem on groups of polynomial growth|Gromov's theorem]], a group of polynomial growth is [[virtually nilpotent]], i.e. it has a [[nilpotent group|nilpotent]] [[subgroup]] of finite [[Index of a subgroup|index]]. In particular, the order of polynomial growth <math>k_0</math> has to be a [[natural numbers|natural number]] and in fact <math>\#(n)\sim n^{k_0}</math>.
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| If <math>\#(n)\ge a^n</math> for some <math>a>1</math> we say that ''G'' has an '''[[exponential growth]] rate'''.
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| Every [[finitely generated group|finitely generated]] ''G'' has at most exponential growth, i.e. for some <math>b>1</math> we have <math>\#(n)\le b^n</math>.
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| If <math>\#(n)</math> grows more slowly than any exponential function, ''G'' has a '''subexponential growth rate'''. Any such group is [[amenable group|amenable]].
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| ==Examples==
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| * A [[free group]] with a finite rank ''k'' > 1 has an exponential growth rate.
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| * A [[finite group]] has constant growth – polynomial growth of order 0 – and includes fundamental groups of manifolds whose universal cover is compact.
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| * If ''M'' is a closed negatively curved [[Riemannian manifold]] then its [[fundamental group]] <math>\pi_1(M)</math> has exponential growth rate. [[John Milnor|Milnor]] proved this using the fact that the [[word metric]] on <math>\pi_1(M)</math> is [[Glossary of Riemannian and metric geometry#Q|quasi-isometric]] to the [[Covering map|universal cover]] of ''M''.
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| * '''Z'''<sup>''d''</sup> has a polynomial growth rate of order ''d''.
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| * The [[discrete Heisenberg group]] ''H''<sub>3</sub> has a polynomial growth rate of order 4. This fact is a special case of the general theorem of [[Hyman Bass|Bass]] and [[Yves Guivarch|Guivarch]] that is discussed in the article on [[Gromov's theorem on groups of polynomial growth|Gromov's theorem]].
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| * The [[lamplighter group]] has an exponential growth. <!-- This is a rare example of a solvable group with exponential growth. -->
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| * The existence of groups with '''intermediate growth''', i.e. subexponential but not polynomial was open for many years. It was asked by [[John Milnor|Milnor]] in 1968 and was finally answered in the positive by [[Rostislav Grigorchuk|Grigorchuk]] in 1984. There are still open questions in this area and a complete picture of which orders of growth are possible and which are not is missing.
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| * The [[triangle group]]s include 3 finite groups (the spherical ones, corresponding to sphere), 3 groups of quadratic growth (the Euclidean ones, corresponding to Euclidean plane), and infinitely many groups of exponential growth (the hyperbolic ones, corresponding to the hyperbolic plane).
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| ==See also==
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| * [[Isoperimetric dimension#Consequences of isoperimetry|Connections to isoperimetric inequalities]]
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| ==References==
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| * [[John Milnor|J. Milnor]], ''A note on curvature and fundamental group'', J. Differential Geometry '''2''' (1968), 1–7.
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| * R. I. Grigorchuk, ''Degrees of growth of finitely generated groups and the theory of invariant means.'', Izv. Akad. Nauk SSSR Ser. Mat. '''48:5''' (1984), 939–985 (Russian).
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| ==Further reading==
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| *{{cite arxiv |author=Rostislav Grigorchuk and [[Igor Pak]] |title=Groups of Intermediate Growth: an Introduction for Beginners |year=2006 |eprint=math.GR/0607384}}
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| [[Category:Infinite group theory]]
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| [[Category:Metric geometry]]
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