- Growth rate (group theory)
In

group theory , the**growth rate**of a group with respect to a symmetricgenerating 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 $x\; in\; T$ then $x^\{-1\}\; in\; T$). Any element $x\; in\; G$ can be expressed as a word in the "T"-alphabet

:$x\; =\; a\_1\; cdot\; a\_2\; cdots\; a\_k\; mbox\{\; where\; \}\; a\_iin\; T$

Let us consider the subset of all elements of "G" which can be presented by such a word of length ≤ "n"

:$B\_n(G,T)\; =\; \{xin\; G\; |\; x\; =\; a\_1\; cdot\; a\_2\; cdots\; a\_k\; mbox\{\; where\; \}\; a\_iin\; T\; mbox\{\; and\; \}\; kle\; n\}$

This set is just the closed ball of radius "n" in the

word metric "d" on "G" with respect to the generating set "T"::$B\_n(G,T)\; =\; \{xin\; G\; |\; d(x,\; e)le\; n\}$

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 thatthey are equivalent ($asim\; b$) if there is a constant "C" such that

:$a(n/\; C)\; leq\; b(n)\; leq\; a(Cn),,$

for example $p^nsim\; q^n$ if $p,q>1$.

Then the growth rate of the group "G" can be defined as the corresponding

equivalence class of the function :$\#(n)=|B\_n(G,T)|,$where $|B\_n(G,T)|$ denotes the number of elements in the set $B\_n(G,T)$.Although the function $\#(n)$ 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 $B\_n(G,T)$ dependon 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 :$\{1over\; C\}\; d\_F(x,y)\; leq\; d\_\{E\}(x,y)\; leq\; C\; d\_F(x,y).$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

:$\#(n)le\; C(n^k+1)$

for some $C,kmath>\; we\; say\; that\; "G"\; has\; a$

**polynomial growth rate**.The infimum $k\_0$ of such "k**s is called the**order of polynomial growth"'.According to Gromov's theorem, a group of polynomial growth is almost nilpotent, i.e. it has a nilpotentsubgroup of finite index. In particular, the order of polynomial growth $k\_0$ has to be a natural number and in fact $\#(n)sim\; n^\{k\_0\}$.If $\#(n)ge\; a^n$ for some $a>1$ we say that "G" has an

. Everyexponential growth ratefinitely generated "G" has at most exponential growth, i.e. for some $b>1$ we have $\#(n)le\; b^n$.If $\#(n)$ grows more slowly than any exponential function, "G" has a

**subexponential growth rate**. Any such group is amenable.**Examples*** A

free group with a finite rank "k" > 1 has an exponential growth rate.* If "M" is a closed negatively curved

Riemannian manifold then itsfundamental group $pi\_1(M)$ has exponential growth rate. Milnor proved this using the fact that theword metric on $pi\_1(M)$ is quasi-isometric to the universal cover of "M".*

**Z**^{"d"}has a polynomial growth rate of order "d".* The

discrete Heisenberg group "H"_{3}has a polynomial growth rate of order 4. This fact is a special case of the general theorem of Bass that is discussed in the article on 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. It was asked by Milnor in 1968 and was finally answered in the positive by 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.**ee also**Connections to isoperimetric inequalities

**References*** J. Milnor, "A note on curvature and fundamental group", J. Differential Geometry

**2**(1968), 1–7.

* 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).

*cite web |author=Rostislav Grigorchuk andIgor Pak |title=Groups of Intermediate Growth: an Introduction for Beginners |year=2006 |publisher=arXiv |url=http://arxiv.org/abs/math.GR/0607384

*Wikimedia Foundation.
2010.*