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Powers in Finitely Generated Groups

Authors:	E Hrushovski P H Kropholler A Lubotzky A Shalev

Institution:	Department of Mathematics, Hebrew University, Jerusalem 91904, Israel P. H. Kropholler ; School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 4NS, United Kingdom A. Lubotzky ; Department of Mathematics, Hebrew University, Jerusalem 91904, Israel A. Shalev ; Department of Mathematics, Hebrew University, Jerusalem 91904, Israel

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 -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.

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