The growth rate of a group is a well-defined notion from asymptotic analysis.
To say that a finitely generated group has polynomial growth means the number of elements of length at most n (relative to a symmetric generating set) is bounded above by a polynomial function p(n).
There is a vast literature on growth rates, leading up to Gromov's theorem.
Yves Guivarc'h[3] and independently Hyman Bass[4] (with different proofs) computed the exact order of polynomial growth.
The Bass–Guivarc'h formula states that the order of polynomial growth of G is where: In particular, Gromov's theorem and the Bass–Guivarc'h formula imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity (excluding for example, fractional powers).
In order to prove this theorem Gromov introduced a convergence for metric spaces.
A relatively simple proof of the theorem was found by Bruce Kleiner.
[5] Later, Terence Tao and Yehuda Shalom modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds.
[6][7] Gromov's theorem also follows from the classification of approximate groups obtained by Breuillard, Green and Tao.
A simple and concise proof based on functional analytic methods is given by Ozawa.
[8] Beyond Gromov's theorem one can ask whether there exists a gap in the growth spectrum for finitely generated group just above polynomial growth, separating virtually nilpotent groups from others.
such that a finitely generated group is virtually nilpotent if and only if its growth function is an
Such a theorem was obtained by Shalom and Tao, with an explicit function