Wilkie, A. J.; van den Dries, Lou An effective bound for groups of linear growth. (English) Zbl 0567.20016 Arch. Math. 42, 391-396 (1984). The main result is an improvement and a completely elementary proof of one special case of M. Gromov’s theorem. Let \(\Gamma\) be a group with a finite generating set X, and put \(G(n)=\#\{g\in G|\) \(g=x_ 1...x_ n\) for certain \(x_ i\in X\cup X^{-1}\cup \{1\}\}\) for each \(n\geq 0\). M. Gromov’s result [Publ. Math., Inst. Hautes Etud. Sci. 53, 53-78 (1981; Zbl 0474.20018)] states that if G(n) is majorized by a polynomial then \(\Gamma\) is nilpotent by finite. It is deep. The authors show that if \(\Gamma\) is infinite, if \(m>0\) and if G(m)-G(m-1)\(\leq m\) then \(\Gamma\) has an infinite cyclic subgroup of index \(\leq m^ 4\). Reviewer: J.D.Macdonald Cited in 17 Documents MSC: 20F05 Generators, relations, and presentations of groups 20E34 General structure theorems for groups 20F19 Generalizations of solvable and nilpotent groups Keywords:growth function; groups of polynomial growth; Gromov’s theorem; nilpotent by finite; infinite cyclic subgroup Citations:Zbl 0474.20018 PDFBibTeX XMLCite \textit{A. J. Wilkie} and \textit{L. van den Dries}, Arch. Math. 42, 391--396 (1984; Zbl 0567.20016) Full Text: DOI References: [1] L.van den Dries and A. J.Wilkie, Gromov’s Theorem on groups of polynomial growth and elementary logic. To appear in J. Algebra. · Zbl 0552.20017 [2] M. Gromov, Groups of polynomial growth and expanding maps. Publ. Math. IHES53, 53-73 (1981). · Zbl 0474.20018 [3] J. Justin, Groupes et semi-groupes à croissance linéaire. C.R. Paris273, 212-214 (1971). · Zbl 0218.20050 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.