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

For g=a1a2amG, aiA, let l(g) be the minimum value of m.




The functionMathworldPlanetmath γ is called the growth function for G with generating set A. If γ is either

(a) bounded above by a polynomial function,

(b) bounded below by an exponential functionDlmfDlmfMathworld, 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
Canonical name Growth
Date of creation 2013-03-22 14:36:09
Last modified on 2013-03-22 14:36:09
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Definition
Classification msc 20F99
Classification msc 20E99
Related topic GrowthOfExponentialFunction
Defines polynomial growth
Defines intermediate growth
Defines exponential growth