Powers in finitely generated groups

E. Hrushovski*, P. H. Kropholler, A. Lubotzky, A. Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


In this paper we study the set Γn of nth-powers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γn contains a finite index subgroup, then Γ is nilpotent-by-finite. We also show that, if Γ is linear and Γn has finite index (i.e. Γ may be covered by finitely many translations of Γn), then Γ 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.

Original languageAmerican English
Pages (from-to)291-304
Number of pages14
JournalTransactions of the American Mathematical Society
Issue number1
StatePublished - 1996


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