In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. 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 then ). Any element can be expressed as a word in the "T"-alphabet
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Let us consider the subset of all elements of "G" which can be presented by such a word of length ≤ "n"
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This set is just the closed ball of radius "n" in the word metric "d" on "G" with respect to the generating set "T":
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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 thatthey 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 dependon 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
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for some