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Title: Co-rank and Betti number of a group (English)
Author: Gelbukh, Irina
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 65
Issue: 2
Year: 2015
Pages: 565-567
Summary lang: English
Category: math
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. (English)
Keyword: co-rank
Keyword: inner rank
Keyword: fundamental group
MSC: 14F35
MSC: 20E05
MSC: 20E06
MSC: 20F34
MSC: 20F99
MSC: 57M07
idZBL: Zbl 06486966
idMR: MR3360446
DOI: 10.1007/s10587-015-0195-0
Date available: 2015-06-16T18:08:07Z
Last updated: 2017-08-07
Stable URL:
Reference: [1] Arnoux, P., Levitt, G.: Sur l'unique ergodicité des 1-formes fermées singulières.Invent. Math. French 84 (1986), 141-156. Zbl 0577.58021, MR 0830042, 10.1007/BF01388736
Reference: [2] Dimca, A., Papadima, S., Suciu, A. I.: Quasi-{K}ähler groups, 3-manifold groups, and formality.Math. Z. 268 (2011), 169-186. Zbl 1228.14018, MR 2805428, 10.1007/s00209-010-0664-y
Reference: [3] Gelbukh, I.: Close cohomologous Morse forms with compact leaves.Czech. Math. J. 63 (2013), 515-528. Zbl 1289.57009, MR 3073975, 10.1007/s10587-013-0034-0
Reference: [4] Gelbukh, I.: The number of split points of a Morse form and the structure of its foliation.Math. Slovaca 63 (2013), 331-348. MR 3037071, 10.2478/s12175-013-0101-x
Reference: [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. Zbl 1274.57005, MR 2654204
Reference: [6] Gelbukh, I.: On the structure of a Morse form foliation.Czech. Math. J. 59 (2009), 207-220. Zbl 1224.57010, MR 2486626, 10.1007/s10587-009-0015-5
Reference: [7] Jaco, W.: Geometric realizations for free quotients.J. Aust. Math. Soc. 14 (1972), 411-418. Zbl 0259.57004, MR 0316571, 10.1017/S1446788700011034
Reference: [8] Leininger, C. J., Reid, A. W.: The co-rank conjecture for 3-manifold groups.Algebr. Geom. Topol. 2 (2002), 37-50. Zbl 0983.57001, MR 1885215, 10.2140/agt.2002.2.37
Reference: [9] Lyndon, R. C., Schupp, P. E.: Combinatorial Group Theory.Classics in Mathematics, Springer Berlin (2001). Zbl 0997.20037, MR 1812024
Reference: [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. Zbl 0511.20019, MR 0682490, 10.1070/IM1983v021n03ABEH001803
Reference: [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. Zbl 0957.57018, MR 1716286
Reference: [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. MR 0755958, 10.1070/IM1985v025n01ABEH001272
Reference: [13] Sikora, A. S.: Cut numbers of 3-manifolds.Trans. Am. Math. Soc. 357 (2005), 2007-2020. Zbl 1064.57018, MR 2115088, 10.1090/S0002-9947-04-03581-0


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