Powers in Finitely Generated Groups

Abstract

In this paper we study the set Γ n \Gamma ^n of n t h n^{th} -powers in certain finitely generated groups Γ \Gamma . We show that, if Γ \Gamma is soluble or linear, and Γ n \Gamma ^n contains a finite index subgroup, then Γ \Gamma is nilpotent-by-finite. We also show that, if Γ \Gamma is linear and Γ n \Gamma ^n has finite index (i.e. Γ \Gamma may be covered by finitely many translations of Γ n \Gamma ^n ), then Γ \Gamma is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the S S -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.