Growth Rate (group Theory) - Definition

Definition

Suppose G is a finitely generated group; and T is a finite symmetric set of generators (symmetric means that if then ). Any element can be expressed as a word in the T-alphabet

Let us consider the subset of all elements of G which can be presented 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 which 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 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 (see below) 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.