Czechoslovak Mathematical Journal

, Volume 65, Issue 2, pp 565–567 | Cite as

Co-rank and Betti number of a group

Article

Abstract

For a finitely generated group, we study the relations between its rank, the maximal rank of its free quotient, called co-rank (inner rank, cut number), and the maximal rank of its free abelian quotient, called the Betti number. We show that any combination of the group’s rank, co-rank, and Betti number within obvious constraints is realized for some finitely presented group (for Betti number equal to rank, the group can be chosen torsion-free). In addition, we show that the Betti number is additive with respect to the free product and the direct product of groups. Our results are important for the theory of foliations and for manifold topology, where the corresponding notions are related with the cut-number (or genus) and the isotropy index of the manifold, as well as with the operations of connected sum and direct product of manifolds.

Keywords

co-rank inner rank fundamental group 

MSC 2010

20E05 20F34 14F35 

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References

  1. [1]
    P. Arnoux, G. Levitt: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math. 84 (1986), 141–156. (In French.)MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    A. Dimca, S. Papadima, A. I. Suciu: Quasi-Kähler groups, 3-manifold groups, and formality. Math. Z. 268 (2011), 169–186.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    I. Gelbukh: Close cohomologous Morse forms with compact leaves. Czech. Math. J. 63 (2013), 515–528.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    I. Gelbukh: The number of split points of a Morse form and the structure of its foliation. Math. Slovaca 63 (2013), 331–348.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    I. Gelbukh: Number of minimal components and homologically independent compact leaves for a Morse form foliation. Stud. Sci. Math. Hung. 46 (2009), 547–557.MATHMathSciNetGoogle Scholar
  6. [6]
    I. Gelbukh: On the structure of a Morse form foliation. Czech. Math. J. 59 (2009), 207–220.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    W. Jaco: Geometric realizations for free quotients. J. Aust. Math. Soc. 14 (1972), 411–418.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    C. J. Leininger, A. W. Reid: The co-rank conjecture for 3-manifold groups. Algebr. Geom. Topol. 2 (2002), 37–50.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    R. C. Lyndon, P. E. Schupp: Combinatorial Group Theory. Classics in Mathematics, Springer, Berlin, 2001.MATHGoogle Scholar
  10. [10]
    G. S. Makanin: Equations in a free group. Math. USSR, Izv. 21 (1983), 483–546; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46 (1982), 1199–1273. (In Russian.)MATHCrossRefGoogle Scholar
  11. [11]
    I. A. Mel’nikova: Maximal isotropic subspaces of skew-symmetric bilinear mapping. Mosc. Univ. Math. Bull. 54 (1999), 1–3; translation from Vestn. Mosk. Univ., Ser I (1999), 3–5. (In Russian.)MathSciNetGoogle Scholar
  12. [12]
    A. A. Razborov: On systems of equations in a free group. Math. USSR, Izv. 25 (1985), 115–162; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48 (1984), 779–832. (In Russian.)MATHCrossRefGoogle Scholar
  13. [13]
    A. S. Sikora: Cut numbers of 3-manifolds. Trans. Am. Math. Soc. 357 (2005), 2007–2020.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2015

Authors and Affiliations

  1. 1.Centro de Investigación en ComputaciónInstituto Politécnico NacionalDF, Mexico CityMexico

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