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
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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 -