Article
MSC:
14F35,
20E05,
20E06,
20F34,
20F99,
57M07 | MR 3360446 | Zbl 06486966 | DOI: 10.1007/s10587-015-0195-0
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Keywords:
co-rank; inner rank; fundamental group
co-rank; inner rank; fundamental group
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.
References:
[1] Arnoux, P., Levitt, G.: Sur l'unique ergodicité des 1-formes fermées singulières. Invent. Math. French 84 (1986), 141-156. DOI 10.1007/BF01388736 | MR 0830042 | Zbl 0577.58021
[2] Dimca, A., Papadima, S., Suciu, A. I.: Quasi-{K}ähler groups, 3-manifold groups, and formality. Math. Z. 268 (2011), 169-186. DOI 10.1007/s00209-010-0664-y | MR 2805428 | Zbl 1228.14018
[3] Gelbukh, I.: Close cohomologous Morse forms with compact leaves. Czech. Math. J. 63 (2013), 515-528. DOI 10.1007/s10587-013-0034-0 | MR 3073975 | Zbl 1289.57009
[4] Gelbukh, I.: The number of split points of a Morse form and the structure of its foliation. Math. Slovaca 63 (2013), 331-348. DOI 10.2478/s12175-013-0101-x | MR 3037071
[5] Gelbukh, I.: Number of minimal components and homologically independent compact leaves for a Morse form foliation. Stud. Sci. Math. Hung. 46 (2009), 547-557. MR 2654204 | Zbl 1274.57005
[6] Gelbukh, I.: On the structure of a Morse form foliation. Czech. Math. J. 59 (2009), 207-220. DOI 10.1007/s10587-009-0015-5 | MR 2486626 | Zbl 1224.57010
[7] Jaco, W.: Geometric realizations for free quotients. J. Aust. Math. Soc. 14 (1972), 411-418. DOI 10.1017/S1446788700011034 | MR 0316571 | Zbl 0259.57004
[8] Leininger, C. J., Reid, A. W.: The co-rank conjecture for 3-manifold groups. Algebr. Geom. Topol. 2 (2002), 37-50. DOI 10.2140/agt.2002.2.37 | MR 1885215 | Zbl 0983.57001
[9] Lyndon, R. C., Schupp, P. E.: Combinatorial Group Theory. Classics in Mathematics, Springer Berlin (2001). MR 1812024 | Zbl 0997.20037
[10] Makanin, G. S.: Equations in a free group. Math. USSR, Izv. 21 (1983), 483-546; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46 1199-1273 (1982), Russian. DOI 10.1070/IM1983v021n03ABEH001803 | MR 0682490 | Zbl 0511.20019
[11] Mel'nikova, I. A.: Maximal isotropic subspaces of skew-symmetric bilinear mapping. Mosc. Univ. Math. Bull. 54 (1999), 1-3; translation from Vestn. Mosk. Univ., Ser I, Russian (1999), 3-5. MR 1716286 | Zbl 0957.57018
[12] Razborov, A. A.: On systems of equations in a free group. Math. USSR, Izv. 25 (1985), 115-162; translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48 779-832 (1984), Russian. DOI 10.1070/IM1985v025n01ABEH001272 | MR 0755958
[13] Sikora, A. S.: Cut numbers of 3-manifolds. Trans. Am. Math. Soc. 357 (2005), 2007-2020. DOI 10.1090/S0002-9947-04-03581-0 | MR 2115088 | Zbl 1064.57018
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