1.3. Gromov’s theorem 21

Let us say that a finite group G is a group of polynomial growth if one

has |B(1,R)| ≤

CRd

for all R ≥ 1 and some constants C, d 0.

Exercise 1.3.2. Show that the notion of a group of polynomial growth (as

well as the rate d of growth) does not depend on the choice of generators

S; thus if S is another set of generators for G, show that G has polynomial

growth with respect to S with rate d if and only if it has polynomial growth

with respect to S with rate d.

Exercise 1.3.3. Let G be a finitely generated group, and let G be a finite

index subgroup of G.

(i) Show that G is also finitely generated. (Hint: Let S be a symmetric

set of generators for G containing the identity, and locate a finite

integer n such that Sn+1G = SnG . Then show that the set S :=

G ∩

S2n+1

is such that

Sn+1

⊂

SnS

. Conclude that

Sn

meets

every coset of S (or equivalently that G =

Sn

S ), and use this

to show that S generates G .)

(ii) Show that G has polynomial growth if and only if G has polynomial

growth.

(iii) More generally, show that any finitely generated subgroup of a

group of polynomial growth also has polynomial growth. Conclude

in particular that a group of polynomial growth cannot contain the

free group on two generators.

From Exercise 1.2.2 we see that the discrete Heisenberg group (1.5) is of

polynomial growth. It is in fact not diﬃcult to show that, more generally,

any nilpotent finitely generated group is of polynomial growth. By Exercise

1.3.3, this implies that any virtually nilpotent finitely generated group is of

polynomial growth.

Gromov’s theorem asserts the converse statement:

Theorem 1.3.1 (Gromov’s theorem [Gr1981]). Let G be an object with

the following properties:

(1) (Group-like object) G is a finitely generated group.

(2) (Weak regularity) G is of polynomial growth.

Then there exists a subgroup G of G such that

(i) (G is close to G) The index |G/G | is finite.

(ii) (Nilpotent type structure) G is nilpotent.

More succinctly: A finitely generated group is of polynomial growth if

and only if it is virtually nilpotent.