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

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

DOI: https://doi.org/10.1090/S0002-9947-96-01456-0

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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.

References

J.-H. Evertse, On equations in $S$-units and the Thue-Mahler equation, Invent. Math. 75 (1984), no. 3, 561–584. MR 735341, DOI 10.1007/BF01388644
J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, On $S$-unit equations in two unknowns, Invent. Math. 92 (1988), no. 3, 461–477. MR 939471, DOI 10.1007/BF01393743
J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, $S$-unit equations and their applications, New advances in transcendence theory (Durham, 1986) Cambridge Univ. Press, Cambridge, 1988, pp. 110–174. MR 971998
J. R. J. Groves, Soluble groups with every proper quotient polycyclic, Illinois J. Math. 22 (1978), no. 1, 90–95. MR 474887
V. S. Guba, A finitely generated complete group, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 5, 883–924 (Russian). MR 873654
P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787–801. MR 105441, DOI 10.1215/ijm/1255448649
P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc. (3) 9 (1959), 595–622. MR 110750, DOI 10.1112/plms/s3-9.4.595

Strong approximation for linear groups in arbitrary characteristic

Serge Lang, Integral points on curves, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 27–43. MR 130219, DOI 10.1007/BF02698777
John C. Lennox and James Wiegold, Converse of a theorem of Mal′cev on nilpotent groups, Math. Z. 139 (1974), 85–86. MR 360833, DOI 10.1007/BF01194147

Homomorphisms onto finite groups

8

R. C. Mason, Diophantine equations over function fields, London Mathematical Society Lecture Note Series, vol. 96, Cambridge University Press, Cambridge, 1984. MR 754559, DOI 10.1017/CBO9780511752490
R. C. Mason, Norm form equations. III. Positive characteristic, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 409–423. MR 830357, DOI 10.1017/S0305004100064355
Madhav V. Nori, On subgroups of $\textrm {GL}_n(\textbf {F}_p)$, Invent. Math. 88 (1987), no. 2, 257–275. MR 880952, DOI 10.1007/BF01388909
Derek J. S. Robinson and John S. Wilson, Soluble groups with many polycyclic quotients, Proc. London Math. Soc. (3) 48 (1984), no. 2, 193–229. MR 729068, DOI 10.1112/plms/s3-48.2.193
T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, vol. 87, Cambridge University Press, Cambridge, 1986. MR 891406, DOI 10.1017/CBO9780511566042
Stan Wagon, The Banach-Tarski paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge, 1985. With a foreword by Jan Mycielski. MR 803509, DOI 10.1017/CBO9780511609596
Boris Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2) 120 (1984), no. 2, 271–315. MR 763908, DOI 10.2307/2006943