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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
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Category: math
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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
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Date available: 2015-06-16T18:08:07Z
Last updated: 2017-08-07
Stable URL: http://hdl.handle.net/10338.dmlcz/144289
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