# growth

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

For $g=a_{1}a_{2}\ldots a_{m}\in G$, $a_{i}\in A$, let $l(g)$ be the minimum value of $m$.

Define

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

.

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

(c) intermediate growth.

For a survey on the topic, see: R. I. Grigorchuk, On growth in group theory, Proceedings of the International 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.

Title growth Growth 2013-03-22 14:36:09 2013-03-22 14:36:09 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 20F99 msc 20E99 GrowthOfExponentialFunction polynomial growth intermediate growth exponential growth