TY - JOUR

T1 - Powers in finitely generated groups

AU - Hrushovski, E.

AU - Kropholler, P. H.

AU - Lubotzky, A.

AU - Shalev, A.

PY - 1996

Y1 - 1996

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=23744503656&partnerID=8YFLogxK

U2 - 10.1090/s0002-9947-96-01456-0

DO - 10.1090/s0002-9947-96-01456-0

M3 - Article

AN - SCOPUS:23744503656

SN - 0002-9947

VL - 348

SP - 291

EP - 304

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 1

ER -