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Powers in Finitely Generated Groups |
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| Authors: | E. Hrushovski P. H. Kropholler A. Lubotzky A. Shalev |
| Affiliation: | 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  of  -powers in certain finitely generated groups  . We show that, if  is soluble or linear, and  contains a finite index subgroup, then  is nilpotent-by-finite. We also show that, if  is linear and  has finite index (i.e.  may be covered by finitely many translations of  ), then  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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