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Co-rank and Betti number of a group. (English) Zbl 1363.20034
Summary: 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.

MSC:
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M07 Topological methods in group theory
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References:
[1] P. Arnoux, G. Levitt: Sur l’unique ergodicité des 1-formes fermées singulières. Invent. Math. 84 (1986), 141–156. (In French.) · Zbl 0577.58021 · doi:10.1007/BF01388736
[2] A. Dimca, S. Papadima, A. I. Suciu: Quasi-Kähler groups, 3-manifold groups, and formality. Math. Z. 268 (2011), 169–186. · Zbl 1228.14018 · doi:10.1007/s00209-010-0664-y
[3] I. Gelbukh: Close cohomologous Morse forms with compact leaves. Czech. Math. J. 63 (2013), 515–528. · Zbl 1289.57009 · doi:10.1007/s10587-013-0034-0
[4] I. Gelbukh: The number of split points of a Morse form and the structure of its foliation. Math. Slovaca 63 (2013), 331–348. · Zbl 1324.57004 · doi:10.2478/s12175-013-0101-x
[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. · Zbl 1274.57005
[6] I. Gelbukh: On the structure of a Morse form foliation. Czech. Math. J. 59 (2009), 207–220. · Zbl 1224.57010 · doi:10.1007/s10587-009-0015-5
[7] W. Jaco: Geometric realizations for free quotients. J. Aust. Math. Soc. 14 (1972), 411–418. · Zbl 0259.57004 · doi:10.1017/S1446788700011034
[8] C. J. Leininger, A. W. Reid: The co-rank conjecture for 3-manifold groups. Algebr. Geom. Topol. 2 (2002), 37–50. · Zbl 0983.57001 · doi:10.2140/agt.2002.2.37
[9] R. C. Lyndon, P. E. Schupp: Combinatorial Group Theory. Classics in Mathematics, Springer, Berlin, 2001.
[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.) · Zbl 0527.20018 · doi:10.1070/IM1983v021n03ABEH001803
[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.)
[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.) · Zbl 0579.20019 · doi:10.1070/IM1985v025n01ABEH001272
[13] A. S. Sikora: Cut numbers of 3-manifolds. Trans. Am. Math. Soc. 357 (2005), 2007–2020. · Zbl 1064.57018 · doi:10.1090/S0002-9947-04-03581-0
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