Geodesic
Co-rank and Betti number of a group
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 565-567
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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.
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.
DOI : 10.1007/s10587-015-0195-0
Classification : 14F35, 20E05, 20E06, 20F34, 20F99, 57M07
Keywords: co-rank; inner rank; fundamental group 
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Gelbukh, Irina. Co-rank and Betti number of a group. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 565-567. doi: 10.1007/s10587-015-0195-0

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