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.
Read more about this topic: Growth Rate (group Theory)
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)