Growth rate (group theory)

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.

Suppose G is a finitely generated group; and T is a finite symmetric set of generators (symmetric means that if ${\displaystyle x\in T}$ then ${\displaystyle x^{-1}\in T}$). Any element ${\displaystyle x\in G}$ can be expressed as a word in the T-alphabet

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