Growth rate (group theory)

In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric 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 generators
(symmetric means that if formula_1 then formula_2).
Any element formula_3 can be expressed as a word in the "T"-alphabet
Consider the subset of all elements of "G" that can be expressed by such a word of length ≤ "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":
More geometrically, formula_7 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 (formula_8) if there is a constant "C" such that for all positive integers "n",
for example formula_10 if formula_11.
Then the growth rate of the group "G" can be defined as the corresponding equivalence class of the function
where formula_13 denotes the number of elements in the set formula_7. Although the function formula_15 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 formula_7 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
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
for some formula_19 we say that "G" has a polynomial growth rate.
The infimum formula_20 of such "ks is called the order of polynomial growth"'.
According to Gromov's theorem, a group of polynomial growth is a virtually nilpotent group, i.e. it has a nilpotent subgroup of finite index. In particular, the order of polynomial growth formula_20 has to be a natural number and in fact formula_22.
If formula_23 for some formula_24 we say that "G" has an exponential growth rate.
Every finitely generated "G" has at most exponential growth, i.e. for some formula_25 we have formula_26.
If formula_15 grows more slowly than any exponential function, "G" has a subexponential growth rate. Any such group is amenable.