Every finitely generated group is either of polynomial growth, intermediate growth, or exponential growth.
As a statement, there is not much to this, the only mathematical content is that the growth function of every finitely generated group has an exponential upper bound.
But as a method of classifying finitely generated groups, it has been very fruitful: Gromov's theorem on groups of polynomial growth; the incredibly rich theory that arose from Grigorchuk's original construction of an intermediate growth group; and the emergence of rich classes of exponential growth groups such as word hyperbolic groups.
