22 1. Introduction

Groups of polynomial growth are related to approximate groups by the

following observation.

Exercise 1.3.4 (Pigeonhole principle). Let G be a finitely generated group

of polynomial growth, and let S be a symmetric set of generators for G

containing the identity.

(i) Show that there exists a C 1 such that |B(1, 5R/2)|≤C|B(1,R/2)|

for a sequence R = Rn of radii going to infinity.

(ii) Show that there exists a K 1 such that B(1,R) is a K-approximate

group for a sequence R = Rn of radii going to infinity. (Hint: Argue

as in Exercise 1.2.1.)

In Chapter 9 we will use this connection to deduce Theorem 1.3.1 from

Theorem 1.2.12. From a historical perspective, this was not the first proof

of Gromov’s theorem; Gromov’s original proof in [Gr1981] relied instead

on a variant of Theorem 1.1.9 (as did some subsequent variants of Gromov’s

argument, such as the nonstandard analysis variant in [vdDrWi1984]), and

a subsequent proof of Kleiner [Kl2010] went by a rather different route,

based on earlier work of Colding and Minicozzi [CoMi1997] on harmonic

functions of polynomial growth. (This latter proof is discussed in [Ta2009,

§1.2] and [Ta2011c, §2.5].) The proof we will give in this text is more recent,

based on an argument of Hrushovski [Hr2012]. We remark that the strategy

used to prove Theorem 1.2.12 — namely taking an ultralimit of a sequence

of approximate groups — also appears in Gromov’s original

argument5.

We

will discuss these sorts of limits more carefully in Chapter 7, but an informal

example to keep in mind for now is the following: If one takes a discrete

group (such as

Zd)

and rescales it (say to

1

N

Zd

for a large parameter N),

then intuitively this rescaled group “converges” to a continuous group (in

this case

Rd).

More generally, one can generate locally compact groups (or

at least locally compact spaces) out of the limits of (suitably normalised)

groups of polynomial growth or approximate groups, which is one of the basic

observations that tie the three different topics discussed above together.

As we shall see in Chapter 10, finitely generated groups arise naturally as

the fundamental groups of compact manifolds. Using the tools of Riemann-

ian geometry (such as the Bishop-Gromov inequality), one can relate the

growth of such groups to the curvature of a metric on such a manifold. As a

consequence, Gromov’s theorem and its variants can lead to some nontrivial

conclusions about the relationship between the topology of a manifold and

its geometry. The following simple consequence is typical:

5Strictly speaking, he uses Gromov-Hausdorff limits instead of ultralimits, but the two types

of limits are closely related, as we shall see in Chapter 7.