20 1. Introduction The proof of this theorem relies on the Gleason-Yamabe theorem (The- orem 1.1.13), and will be discussed in Chapter 8. The key connection will take some time to explain properly, but roughly speaking, it comes from the fact that the ultraproduct of a sequence of K-approximate groups can be used to generate a locally compact group, to which the Gleason-Yamabe theorem can be applied. This in turn can be used to place a metric on ap- proximate groups that obeys a commutator estimate similar to (1.2), which allows one to run an argument similar to that used to prove Theorem 1.0.2. 1.3. Gromov’s theorem The final topic of this chapter will be Gromov’s theorem on groups of poly- nomial growth [Gr1981]. This theorem is analogous to Theorem 1.1.13 or Theorem 1.2.12, but in the category of finitely generated groups rather than locally compact groups or approximate groups. Let G be a group that is generated by a finite set S of generators for notational simplicity we will assume that S is symmetric and contains the origin. Then S defines a (right-invariant) word metric on G, defined by setting d(x, y) for x, y ∈ G to be the least natural number n such that x ∈ Sny. One easily verifies that this is indeed a metric that is right-invariant (thus d(xg, yg) = d(x, y) for all x, y, g ∈ G). Geometrically, this metric describes the geometry of the Cayley graph on G formed by connecting x to sx for each x ∈ G and s ∈ S. (See [Ta2011c, §2.3] for more discussion of using Cayley graphs to study groups geometrically.) Let us now consider the growth of the balls B(1,R) = S R as R → ∞, where R is the integer part of R. On the one hand, we have the trivial upper bound |B(1,R)| ≤ |S|R that shows that such balls can grow at most exponentially. And for “typical” nonabelian groups, this exponential growth actually occurs consider the case for instance when S consists of the generators of a free group (together with their inverses, and the group identity). However, there are some groups for which the balls grow at a much slower rate. A somewhat trivial example is that of a finite group G, since clearly |B(1,R)| will top out at |G| (when R reaches the diameter of the Cayley graph) and stop growing after that point. Another key example is the abelian case: Exercise 1.3.1. If G is an abelian group generated by a finite symmetric set S containing the identity, show that |B(1,R)| ≤ (1 + R)|S|. In particular, B(1,R) grows at a polynomial rate in R.

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