Co-rank and Betti number of a group

Gelbukh, Irina. "Co-rank and Betti number of a group." Czechoslovak Mathematical Journal 65.2 (2015): 565-567. <http://eudml.org/doc/270107>.

@article{Gelbukh2015,
abstract = {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.},
author = {Gelbukh, Irina},
journal = {Czechoslovak Mathematical Journal},
keywords = {co-rank; inner rank; fundamental group},
language = {eng},
number = {2},
pages = {565-567},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Co-rank and Betti number of a group},
url = {http://eudml.org/doc/270107},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Gelbukh, Irina
TI - Co-rank and Betti number of a group
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 565
EP - 567
AB - 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.
LA - eng
KW - co-rank; inner rank; fundamental group
UR - http://eudml.org/doc/270107
ER -