Stable exponents in discrete groups.
type: Item
instance of: scholarly article
title: Stable exponents in discrete groups.
zbMATH Open document ID: 1059.20037
DOI: 10.1023/A:1021227311187
author: Michel Coornaert
published in: Geometriae Dedicata
publication date: 2003-03-11T00:00:00Z
review text: Let \(G\) be an arbitrary group. The stable exponent \(p_{+}(g)\) of an element \(g \in G\) is defined as \[ p_{+}(g)=\mathop{\lim \sup}_{n \to \infty} \frac{p(g^n)}{n}, \] where \(p(g^n)\) denotes the supremum of the set of integers \(k \geq 1\) such that \(g^n\) admits a \(k\)th root in \(G\). \textit{M.Gromov} [In: Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 75-263 (1987; Zbl 0634.20015)] has observed that in a word hyperbolic group the stable exponent of every non-torsion element is an integer. The author of the paper proves that this is also true for non-torsion element in finitely generated nilpotent groups. Besides this result, the author shows that, for any rational number \(r \geq 1\), there exists a torsion-free CAT(0) group containing an element whose stable exponent is equal to \(r\).
reviewed by: N. Yu. Makarenko
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1879685
zbMATH Keywords: stable exponents
zbMATH Keywords: finitely generated nilpotent groups
zbMATH Keywords: CAT(0)-groups
MaRDI profile type: Publication
full work available at URL: a:1021227311187
OpenAlex ID: W3203274
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description: scientific article; zbMATH DE number 1879685