Editing Growth rate (group theory)

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In the mathematical subject of [[geometric group theory]], the '''growth rate''' of a [[group (mathematics)|group]] with respect to a symmetric [[generating set of a group|generating set]] describes how fast a group grows.  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 [[Generating set of a group|generator]]s
(symmetric means that if <math> x \in T </math> then <math> x^{-1} \in T </math>).
Any element <math> x \in G </math> can be expressed as a [[string (computer science)#Formal theory|word]] in the ''T''-alphabet

:<math> x = a_1 \cdot a_2 \cdots a_k \text{ where } a_i\in T. </math>

Consider the subset of all elements of ''G'' that can be expressed by such a word of length &le;&nbsp;''n''

:<math>B_n(G,T) = \{x\in G \mid x = a_1 \cdot a_2 \cdots a_k \text{ where } a_i\in T \text{ and } k\le n\}.</math>

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'':

:<math>B_n(G,T) = \{x\in G \mid d(x, e)\le n\}.</math>

More geometrically, <math>B_n(G,T)</math> is the set of vertices in the [[Cayley graph]] with respect to ''T'' that are within distance ''n'' of the identity.

Given two nondecreasing positive functions ''a'' and ''b'' one can say that they are equivalent (<math>a\sim b</math>) if there is a constant ''C'' such that for all positive integers&nbsp;''n'',

:<math> a(n/ C) \leq b(n) \leq a(Cn),\, </math>

for example <math> p^n\sim q^n </math> if <math> p,q>1 </math>.

Then the growth rate of the group ''G'' can be defined as the corresponding [[equivalence class]] of the function
:<math>\#(n)=|B_n(G,T)|, </math>
where <math>|B_n(G,T)|</math> denotes the number of elements in the set <math>B_n(G,T)</math>. Although the function <math>\#(n)</math> 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 <math>B_n(G,T)</math> depend 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
:<math> {1\over C} \ d_F(x,y) \leq d_E(x,y) \leq C \ d_F(x,y). </math>
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==<!-- This section is linked from [[Hyperbolic geometry]] -->

If

:<math>\#(n)\le C(n^k+1)</math>

for some <math>C,k<\infty</math> we say that ''G'' has a '''polynomial growth rate'''.
The infimum <math>k_0</math> of such ''k'''s is called the '''order of polynomial growth'''.
According to [[Gromov's theorem on groups of polynomial growth|Gromov's theorem]], a group of polynomial growth is a [[virtually]] [[nilpotent group]], 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>.

If <math>\#(n)\ge a^n</math> for some <math>a>1</math> we say that ''G'' has an '''[[exponential growth]] rate'''.
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>.

If <math>\#(n)</math> grows [[Infra-exponential|more slowly than any exponential function]], ''G'' has a '''subexponential growth rate'''. Any such group is [[amenable group|amenable]].

==Examples==

* A [[free group]] of finite rank <math>k > 1</math> has exponential growth rate.
* A [[finite group]] has constant growth—that is, polynomial growth of order 0—and this includes [[fundamental group]]s of [[manifold]]s whose [[universal cover]] is [[compact space|compact]].
* If ''M'' is a [[closed manifold|closed]] [[Curvature of Riemannian manifolds|negatively curved]] [[Riemannian manifold]] then its [[fundamental group]] <math>\pi_1(M)</math> has exponential growth rate. [[John 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''.
* The [[free abelian group]] <math>\Z^d</math> has a polynomial growth rate of order ''d''.
* The [[discrete Heisenberg group]] <math>H_3</math> has a polynomial growth rate of order 4. This fact is a special case of the general theorem of [[Hyman Bass]] and [[Yves Guivarch]] that is discussed in the article on [[Gromov's theorem on groups of polynomial growth|Gromov's theorem]].
* The [[lamplighter group]] has an exponential growth.  <!-- This is a rare example of a solvable group with exponential growth.  -->
* The existence of groups with '''intermediate growth''', i.e. subexponential but not polynomial was open for many years. The question was asked by Milnor in 1968 and was finally answered in the positive by [[Rostislav 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.
* The [[triangle group]]s include infinitely many finite groups (the spherical ones, corresponding to sphere), three 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).

==See also==
* [[Isoperimetric dimension#Consequences of isoperimetry|Connections to isoperimetric inequalities]]

==References==

* {{cite journal | author = Milnor J. | author-link = John Milnor | year = 1968 | title = A note on curvature and fundamental group | journal =  	Journal of Differential Geometry | volume = 2 | pages = 1–7 | doi=10.4310/jdg/1214501132| doi-access = free }}
* {{cite journal | author = Grigorchuk R. I. | year = 1984 | title = Degrees of growth of finitely generated groups and the theory of invariant means. | journal = Izv. Akad. Nauk SSSR Ser. Mat. | volume = 48 | issue = 5| pages = 939–985 | language=ru}}

==Further reading==
*{{cite arXiv |author=Rostislav Grigorchuk and [[Igor Pak]] |title=Groups of Intermediate Growth: an Introduction for Beginners |year=2006 |eprint=math.GR/0607384}}

[[Category:Infinite group theory]]
[[Category:Cayley graphs]]
[[Category:Metric geometry]]