Gromov's theorem on groups of polynomial growth

Remarks

The direction "virtually nilpotent implies polynomial growth" was known before Gromov. For a finitely generated nilpotent group the degree of growth is the integer $\sum_{k \ge 1} k \, r_k$. Here $r_k$ is the torsion-free rank of the quotient $\gamma_k(G) / \gamma_{k+1}(G)$ of consecutive terms of the lower central series. Gromov proved the hard direction. His idea is to rescale the word metric of $G$ by $1/n$ and pass to a limit space as $n \to \infty$. Polynomial growth makes this limit a finite-dimensional, locally compact space. The solution of Hilbert's fifth problem turns its isometry group into a Lie group, on which $G$ acts. From this action one extracts the nilpotent subgroup by induction on the degree. Kleiner later gave a different proof based on harmonic functions on the Cayley graph.

Example: the integer Heisenberg group of upper unitriangular $3 \times 3$ integer matrices is nilpotent, with $r_1 = 2$ and $r_2 = 1$, so its growth is polynomial of degree $2 + 2 = 4$.

A consequence is a gap: the growth of a finitely generated group is either polynomial of an integer degree, or faster than every polynomial. Groups of intermediate growth exist, such as the Grigorchuk group, so "faster than every polynomial" does not mean exponential.

Going deeper

For a student

The theorem shows that being virtually nilpotent is a quasi-isometry invariant of finitely generated groups, because growth is. For finitely generated linear groups the gap is sharper: by the Tits alternative and the Milnor–Wolf theorem for solvable groups, their growth is either polynomial or exponential. It is also used in the study of random walks and of manifolds with almost nonnegative curvature.