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Concept
Growth rate (group theory)
Summary
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.
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
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 (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.
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 .
Official source
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