Growth rate (group theory)

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In group theory, the growth rate of a group with respect to a symmetric generating set is a notion that describes how fast a group grows. Building it gradually outwards from the identity element by multiplication by members of the generating set, one can count the number of elements constructed at stage N.

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Definition

Suppose G is a finitely generated group; and T is a finite symmetric set of generators (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 "word" in T-alphabet

<math> x = a_1 \cdot a_2 \cdot

\ldots \cdot a_k \mbox{ where } a_i\in T <math>

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

<math>B_n(G,T) = \{x\in G | x = a_1 \cdot a_2 \cdot

\ldots \cdot a_k \mbox{ where } a_i\in T \mbox{ and } k\le n\},<math>

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

<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 correspondent equivalence class of 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.

A more geometrical definition use the Cayley graph associated to the set T. The described set <math>B_n(G,T)<math> is simply n-ball with the center at identity element e in the Cayley graph metric d.

<math>B_n(G,T) = \{x\in G | d(x, e)\le n\},<math>

The distance function d and therefore sets <math>B_n(G,T)<math> depend on the generating set. However, any two such metrics are equivalent in the following sense: for finite symmetric generating sets E, F, there are 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 choice of generating set.

Polynomial and exponential growth

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 order of polynomial growth. According to Gromov's theorem a group of polynomial growth is almost nilpotent, i.e. it has nilpotent subgroup of finite index, in particular the order of polynomial growth <math>k_0<math> has to be natural 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 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 slower than exponent it has subexponential growth rate. Any such group is amenable.

Examples

A free group with a finite rank k > 1 has an exponential growth rate.

Z2 has a polynomial growth rate of order 2.

The discrete Heisenberg group H3 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 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.

See also

Connections to isoperimetric inequalities

References

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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