Powers in Finitely Generated Groups

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.