Growth rate of a group

Summary

The growth function of a finitely generated group counts how many elements can be written as words of length at most $n$ in the generators. Its rate of increase, up to a natural equivalence, does not depend on the generating set, and it is a quasi-isometry invariant. Groups split into groups of polynomial growth, of exponential growth, and of intermediate growth. Gromov's theorem identifies the groups of polynomial growth as the virtually nilpotent groups.