Powers in Finitely Generated Groups

Hrushovski, E.; Kropholler, P.; Lubotzky, A.; Shalev, A.

doi:10.1090/S0002-9947-96-01456-0

Powers in Finitely Generated Groups
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by E. Hrushovski, P. H. Kropholler, A. Lubotzky and A. Shalev PDF

Trans. Amer. Math. Soc. 348 (1996), 291-304 Request permission

Abstract:

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

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