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
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
and rescales it (say to
for a large parameter N),
then intuitively this rescaled group “converges” to a continuous group (in
this case
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.
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