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.

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