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)


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 


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


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  1. [1]
    H. Bass, M. Lazard, J.-P. Serre. Sous-groupes d’indice fini dans SL(n, ℤ, Bull. Amer. Math. Soc., 70 (1964), 385–392.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge: Cambridge University Press, 2008.Google Scholar
  3. [3]
    O. Bogopolski, Arboreal decomposability of groups of automorphisms of free groups, Algebra and Logic, 26, no. 2 (1987), 79–91.CrossRefGoogle Scholar
  4. [4]
    O. Bogopolski, Classification of automorphisms of the free group of rank 2 by ranks of fixed-points subgroups, J. Group Theory, 3, no. 3 (2000), 339–351.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    F. Grunewald, A. Lubotzky, Linear representations of the automorphism group of a free group, GAFA, 18, no. 5 (2009), 1564–1608.CrossRefMathSciNetGoogle Scholar
  6. [6]
    P. de la Harpe, A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175, 1989.Google Scholar
  7. [7]
    S. Hoory, N. Linial, A. Wigderson, Expander graphs and their applications, Bulletin of the Amer. Math. Soc., 43, no. 4 (2006), 439–561.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    D. Kazhdan, On the connection of the dual space of a group with the structure of its closed subgroups, Functional analysis and its applications, 1 (1967), 63–65.zbMATHCrossRefGoogle Scholar
  9. [9]
    A. Lubotzky, Discrete groups, expanding graphs and invariant measures, Basel-Boston-Berlin: Birkhäuser, 1994.zbMATHGoogle Scholar
  10. [10]
    A. Lubotzky, I. Pak, The product replacement algorithm and Kazhdan’s property (T), J. Amer. Math. Soc., 14, no. 2 (2001), 347–363.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    R.C. Lyndon, P.E. Schupp, Combinatorial group theory, Berlin: Springer-Verlag, 1977.zbMATHGoogle Scholar
  12. [12]
    W. Magnus, Über n-dimensionale Gittertransformationen, Acta Math., 64 (1935), 353–367zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    W. Magnus, A. Karrass, D. Solitar, Combinatorial group theory, New York: Wiley, 1966.zbMATHGoogle Scholar
  14. [14]
    G.A. Margulis, Explicit constructions of concentrators. Problems of Inform. Transm., 10 (1975), 325–332. [Russian original: Problemy Peredatci Informacii, 9 (1973), 71-80.]Google Scholar
  15. [15]
    J. McCool, A faithful polynomial presentation of Out(F 3), Math. Proc. Camb. Phil. Soc., 106, no. 2 (1989), 207–213.zbMATHMathSciNetGoogle Scholar
  16. [16]
    J. Mennicke, Finite factor groups of the unimodular group, Ann. Math., Ser 2., 81 (1965), 31–37.CrossRefMathSciNetGoogle Scholar
  17. [17]
    J. Nielsen, Die Gruppe der dreidimensionalen Gittertransformationen, Danske Vid. Selsk. Mat.-Fys. Medd. 12 (1924), 1–29.Google Scholar
  18. [18]
    J. Nielsen, Die Isomorphismengruppe der freien Gruppen, Math. Ann, 91 (1924), 169–209.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    T. Satoh, The abelianization of the congruence IA-automorphism group of a free group, Math. Proc. Camb. Philos. Soc., 142, no. 2 (2007), 239–248.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    J.-P. Serre, Trees. Berlin-Heidelberg-New York: Springer-Verlag, 1980.zbMATHGoogle Scholar
  21. [21]
    Y. Shalom, The algebraization of Kazhdan’s property (T). Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zurich: European Mathematical Society (EMS). 1283–1310 (2006).Google Scholar
  22. [22]
    B. Sury, T.N. Venkataramana, Generators for all principal congruence subgroups of SLn(ℤ) with n > 2, Proc. Amer. Math. Soc., 122, no. 2 (1994), 355–358.zbMATHMathSciNetGoogle Scholar
  23. [23]
    K. Vogtmann, Automorphisms of free groups and outer space, Geometriae Dedicata, 94 (2002), 1–31.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Y. Watatani, Property (T) of Kazhdan implies property (FA) of Serre, Math. Japon., 27, no. 1 (1982), 97–103.zbMATHMathSciNetGoogle Scholar

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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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