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A celebrated theorem of Gromov reads:
Theorem 1 Every finitely generated group of polynomial growth is virtually nilpotent.
The original proof of Gromov’s theorem was quite non-elementary, using an infinitary limit and exploiting the work surrounding the solution to Hilbert’s fifth problem. More recently, Kleiner provided a proof which was more elementary (based in large part on an earlier paper of Colding and Minicozzi), though still not entirely so, relying in part on (a weak form of the) Tits alternative and also on an ultrafilter argument of Korevaar-Schoen and Mok. I discuss Kleiner’s argument more in this previous blog post.
Recently, Yehuda Shalom and I established a quantitative version of Gromov’s theorem by making every component of Kleiner’s argument finitary. Technically, this provides a fully elementary proof of Gromov’s theorem (we do use one infinitary limit to simplify the argument a little bit, but this is not truly necessary); however, because we were trying to quantify as much of the result as possible, the argument became quite lengthy.
In this note I want to record a short version of the argument of Yehuda and myself which is not quantitative, but gives a self-contained and largely elementary proof of Gromov’s theorem. The argument is not too far from the Kleiner argument, but has a number of simplifications at various places. In a number of places, there was a choice to take between a short argument that was “inefficient” in the sense that it did not lead to a good quantitative bound, and a lengthier argument which led to better quantitative bounds. I have opted for the former in all such cases.
Yehuda and I plan to write a short paper containing this argument as well as some additional material, but due to some interest in this particular proof, we are detailing it here on this blog in advance of our paper.
Note: this post will assume familiarity with the basic terminology of group theory, and will move somewhat quickly through the technical details.
Yehuda Shalom and I have just uploaded to the arXiv our paper “A finitary version of Gromov’s polynomial growth theorem“, to be submitted to Geom. Func. Anal.. The purpose of this paper is to establish a quantitative version of Gromov’s polynomial growth theorem which, among other things, is meaningful for finite groups. Here is a statement of Gromov’s theorem:
Gromov’s theorem. Let be a group generated by a finite (symmetric) set , and suppose that one has the polynomial growth condition
for all sufficiently large and some fixed , where is the ball of radius generated by (i.e. the set of all words in of length at most , evaluated in ). Then is virtually nilpotent, i.e. it has a finite index subgroup which is nilpotent of some finite step .
As currently stated, Gromov’s theorem is qualitative rather than quantitative; it does not specify any relationship between the input data (the growth exponent and the range of scales for which one has (1)), and the output parameters (in particular, the index of the nilpotent subgroup of , and the step of that subgroup). However, a compactness argument (sketched in this previous blog post) shows that some such relationship must exist; indeed, if one has (1) for all for some sufficiently large , then one can ensure has index at most and step at most for some quantities , ; thus Gromov’s theorem is inherently a “local” result which only requires one to multiply the generator set a finite number of times before one sees the virtual nilpotency of the group. However, the compactness argument does not give an explicit value to the quantities , and the nature of Gromov’s proof (using, in particular, the deep Montgomery-Zippin-Yamabe theory on Hilbert’s fifth problem) does not easily allow such an explicit value to be extracted.
Another point is that the original formulation of Gromov’s theorem required the polynomial bound (1) at all sufficiently large scales . A later proof of this theorem by van den Dries and Wilkie relaxed this hypothesis to requiring (1) just for infinitely many scales ; the later proof by Kleiner (which I blogged about here) also has this relaxed hypothesis.
Our main result reduces the hypothesis (1) to a single large scale, and makes most of the qualitative dependencies in the theorem quantitative:
Theorem 1. If (1) holds for some for some sufficiently large absolute constant , then contains a finite index subgroup which is nilpotent of step at most .
The argument does in principle provide a bound on the index of in , but it is very poor (of Ackermann type). If instead one is willing to relax “nilpotent” to “polycyclic“, the bounds on the index are somewhat better (of tower exponential type), though still far from ideal.
There is a related finitary analogue of Gromov’s theorem by Makarychev and Lee, which asserts that any finite group of uniformly polynomial growth has a subgroup with a large abelianisation. The quantitative bounds in that result are quite strong, but on the other hand the hypothesis is also strong (it requires upper and lower bounds of the form (1) at all scales) and the conclusion is a bit weaker than virtual nilpotency. The argument is based on a modification of Kleiner’s proof.
Our argument also proceeds by modifying Kleiner’s proof of Gromov’s theorem (a significant fraction of which was already quantitative), and carefully removing all of the steps which require one to take an asymptotic limit. To ease this task, we look for the most elementary arguments available for each step of the proof (thus consciously avoiding powerful tools such as the Tits alternative). A key technical issue is that because there is only a single scale for which one has polynomial growth, one has to work at scales significantly less than in order to have any chance of keeping control of the various groups and other objects being generated.
Below the fold, I discuss a stripped down version of Kleiner’s argument, and then how we convert it to a fully finitary argument.