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Subgroups of Small Index in Aut(Fn) and Kazhdan’s Property (T)

  • O. Bogopolski
  • R. Vikentiev
Part of the Trends in Mathematics book series (TM)

Abstract

We introduce a series of interesting subgroups of finite index in Aut(F n ). One of them has index 42 in Aut(F 3) and infinite abelianization. This implies that Aut(F 3) does not have Kazhdan’s property (T); see [15] and [5] for other proofs. We prove also that every subgroup of finite index in Aut(F n ), n ⩾ 3, which contains the subgroup of IA-automorphisms has a finite abelianization.

We introduce a subgroup K(n) of finite index in Aut(F n ) and show, that its abelianization is infinite for n=3, and it is finite for n ⩾ 4. We ask, whether the abelianization of its commutator subgroup K(n)′ is infinite for n ⩾ 4. If so, then Aut(F n ) would not have Kazhdan’s property (T) for n ⩾ 4.

Mathematics Subject Classification (2000)

20F28 20E05 20E15 

Keywords

Automorphisms free groups Kazhdan’s property (T) congruence subgroups 

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© Springer Basel AG 2010

Authors and Affiliations

  • O. Bogopolski
    • 1
    • 2
  • R. Vikentiev
    • 1
    • 3
  1. 1.Institute of Mathematics of Siberian Branch of Russian Academy of SciencesNovosibirskRussia
  2. 2.University of DüsseldorfGermany
  3. 3.Novosibirsk State UniversityNovosibirskRussia

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