2011
DOI: 10.1016/j.jpaa.2010.11.017
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Small conjugacy classes in the automorphism groups of relatively free groups

Abstract: a b s t r a c tLet F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F ′ of F and (b) the quotient group F /R is residually torsion-free nilpotent. Then the automorphism group Aut(F /R ′ ) of the group F /R ′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generat… Show more

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Cited by 5 publications
(3 citation statements)
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References 16 publications
(22 reference statements)
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“…Proof. The result is proved in [14] for infinitely generated relatively free groups, and in fact the proof given in [14] can be used without significant changes in the general case. For the reader's convenience, we reproduce the plan of the proof.…”
Section: The Small Index Property For Relatively Free Algebrasmentioning
confidence: 91%
“…Proof. The result is proved in [14] for infinitely generated relatively free groups, and in fact the proof given in [14] can be used without significant changes in the general case. For the reader's convenience, we reproduce the plan of the proof.…”
Section: The Small Index Property For Relatively Free Algebrasmentioning
confidence: 91%
“…Theorem 9 (Tolstykh [21]). Let F ∞ be an infinitely generated free group, R ≤ F ′ ∞ a fully characteristic subgroup of F ∞ such that the quotient group F ∞ /R is residually torsion-free nilpotent.…”
Section: Other Propertiesmentioning
confidence: 99%
“…As to infinite groups, typical examples of complete groups arise as groups of automorphisms G = Aut (F ), where F is a free group (or a group that is "not so far" from free). The cases where F is a free group, free nilpotent group of class two, or the quotient of a free group by an appropriate characteristic subgroup were treated, respectively, in [DF1]- [DF3] (for groups of finite rank) and [To1]- [To3] (for groups of infinite rank). The cases F = GL(n, Z) (n odd), F = PGL(2, Z) (n ≥ 2), F = SL(n, Z) (n ≥ 3) were considered in [Dy].…”
Section: 2mentioning
confidence: 99%