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A finitary version of Gromov’s polynomial growth theorem. (English) Zbl 1262.20044
Let \(G\) be a finitely generated group with fixed finite symmetric generating set \(S\). The set \(B_S(R)\) is \(R\)-ball centered at the identity in the \(S\)-word metric. M. Gromov [Publ. Math., Inst. Hautes Étud. Sci. 53, 53-78 (1981; Zbl 0474.20018)] showed that if \(|B_S(R)|\leq R^d\) for some fixed \(d\) and all sufficiently large \(R\), then \(G\) is virtually nilpotent. This result was later generalized by L. van den Dries and A. J. Wilkie [J. Algebra 89, 349-374 (1984; Zbl 0552.20017)] who proved the result assuming only that the condition \(B_S(R)\leq R^d\) holds at infinitely many scales (instead of all scales). Later, B. Kleiner [J. Am. Math. Soc. 23, No. 3, 815-829 (2010; Zbl 1246.20038)] used different techniques to prove van den Dries’ and Wilkie’s result. The paper under review is a significant strengthening of this theorem that requires the growth condition to be satisfied at only one scale.
More specifically, the authors show that for some explicit constant \(C\) and for every finitely generated group \(G\) and \(d>0\) if there exists an \(R_0>\exp(\exp(Cd^C))\) for which the number of elements in a ball of radius \(R_0\) is bounded by \(R_0^d\), then \(G\) is virtually nilpotent. They also give a bound on the nilpotency degree. This remarkable result also yields a new result for finite groups. Additionally, they show that groups with slightly super-polynomial growth (where the bound is \(R^{c(\log\log R)^c}\)) are virtually nilpotent. Another interesting corollary can be expressed by saying that polynomial growth at one scale implies polynomial growth at all scales.
The proof of the main theorem follows the strategy used by Kleiner, but each step of the proof requires a great deal of work; it is not simply a rehashing of Kleiner’s proof. The authors give a comprehensive introduction that includes a plan of the proof, examples and comparisons with other work as well as all necessary definitions.

MSC:
20F65 Geometric group theory
20F19 Generalizations of solvable and nilpotent groups
20F69 Asymptotic properties of groups
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
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