Gromov's theorem on groups of polynomial growth and elementary logic

In the fall of 1980 the authors attended Professor Tits’ course at Yale University in which he gave an account of Gromov’s beautiful proof that every finitely generated group of polynomial growth has a nilpotent subgroup of finite index. An essential part of Gromov’s argument consists of constructing for each group of polynomial growth a locally compact metric space and an action of a subgroup of finite index on that space. The intuitive motivation underlying this construction is fairly clear but it required an elaborate theory of “limits” of metric spaces to be carried out. It occurred to us to give a simple nonstandard definition of a space which has all the nice properties needed in the rest of Gromov’s argument. Besides shortening proofs our construction works for arbitrary finitely generated groups, not only for those of polynomial growth, and it has functorial properties. This enables us to state some of Gromov’s lemmas without the restriction of polynomial growth, e.g., (4.2) and (5.5). We also found a new proof of local compactness of the space, see Section 6, under an a priori weaker hypothesis than polynomial growth, and this led to a slight extension of Gromov’s theorem: If‘ the group r with finite generating set X has growth function G, with G,(n) < c . nd for infinitely many n and positive constants c, d, then r has a nilpotent subgroup of jinite index. (Gromov’s hypothesis is that G,(n) < c . nd for afl n > 0.)