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An effective bound for groups of linear growth. (English) Zbl 0567.20016

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

MSC:

20F05 Generators, relations, and presentations of groups
20E34 General structure theorems for groups
20F19 Generalizations of solvable and nilpotent groups

Citations:

Zbl 0474.20018
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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
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