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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Powers in Finitely Generated Groups


Authors: E. Hrushovski, P. H. Kropholler, A. Lubotzky and A. Shalev
Journal: Trans. Amer. Math. Soc. 348 (1996), 291-304
MSC (1991): Primary 20G15, 20F16; Secondary 11D99, 20G40, 43A05
MathSciNet review: 1316851
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Abstract: In this paper we study the set $\G^n$ of $n^{th}$-powers in certain finitely generated groups $\G$. We show that, if $\G$ is soluble or linear, and $\G^n$ contains a finite index subgroup, then $\G$ is nilpotent-by-finite. We also show that, if $\G$ is linear and $\G^n$ has finite index (i.e. $\G$ may be covered by finitely many translations of $\G^n$), then $\G$ is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the $S$-unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.


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Additional Information

E. Hrushovski
Affiliation: Department of Mathematics, Hebrew University, Jerusalem 91904, Israel

P. H. Kropholler
Affiliation: School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 4NS, United Kingdom

A. Lubotzky
Affiliation: Department of Mathematics, Hebrew University, Jerusalem 91904, Israel

A. Shalev
Affiliation: Department of Mathematics, Hebrew University, Jerusalem 91904, Israel

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01456-0
PII: S 0002-9947(96)01456-0
Received by editor(s): January 20, 1995
Article copyright: © Copyright 1996 American Mathematical Society