growth

Let $G$ be a finitely generated group with generating set $A$ (closed under inverses).

For $g=a_1{a}_2\mathrm{\dots }{a}_m\in G$, $a_i\in A$, let $l(g)$ be the minimum value of $m$.

Define

$$\gamma (n)=\mid \{g\in G:l(g)\le n\}\mid$$

.

The function $\gamma$ is called the growth function for $G$ with generating set $A$. If $\gamma$ is either

(a) bounded above by a polynomial function,

(b) bounded below by an exponential function, or

(c) neither,

then this condition is preserved under changing the generating set for $G$. Respectively, then, $G$ is said to have

(a) polynomial growth,

(b) exponential growth, or

(c) intermediate growth.

For a survey on the topic, see:R. I. Grigorchuk, On growth in group theory, Proceedings of theInternational Congress of Mathematicians, Kyoto 1990, Volume I,II (Math. Soc. Japan, 1991), pages 325 to 338.

Note that, as the generating set is assumed to be closed under inverses, we need only have $G$ as a semigroup - as such, the above applies equally well in semigroup theory.