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
.
Any element 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, 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 if there is a constant C such that for all positive integers n,
for example if.
Then the growth rate of the group G can be defined as the corresponding equivalence class of the function
where denotes the number of elements in the set. Although the function depends on the set of generators T its rate of growth does not and therefore the rate of growth gives an invariant of a group.
The word metric d and therefore sets 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 we say that G has a polynomial growth rate.
The infimum 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 has to be a natural number and in fact.
If for some we say that G has an exponential growth rate.
Every finitely generated G has at most exponential growth, i.e. for some we have.
If grows more slowly than any exponential function, G has a subexponential growth rate'. Any such group is amenable.

Examples