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Co-rank and Betti number of a group

Irina Gelbukh (2015)

Czechoslovak Mathematical Journal

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...

Correction for the paper “ S 3 -bundles and exotic actions”

T. E. Barros (2001)

Bulletin de la Société Mathématique de France

In [R] explicit representatives for S 3 -principal bundles over S 7 are constructed, based on these constructions explicit free S 3 -actions on the total spaces are described, with quotients exotic 7 -spheres. To describe these actions a classification formula for the bundles is used. This formula is not correct. In Theorem 1 below, we correct the classification formula and in Theorem 2 we exhibit the correct indices of the exotic 7 -spheres that occur as quotients of the free S 3 -actions described above.

Correspondence between diffeomorphism groups and singular foliations

Tomasz Rybicki (2012)

Annales Polonici Mathematici

It is well-known that any isotopically connected diffeomorphism group G of a manifold determines a unique singular foliation ℱ G . A one-to-one correspondence between the class of singular foliations and a subclass of diffeomorphism groups is established. As an illustration of this correspondence it is shown that the commutator subgroup [G,G] of an isotopically connected, factorizable and non-fixing C r diffeomorphism group G is simple iff the foliation ℱ [ G , G ] defined by [G,G] admits no proper minimal sets....

Counting fixed points of a finitely generated subgroup of Aff [C].

F. Loray, M. Van Der Put, F. Recher (2004)

Publicacions Matemàtiques

Given a finitely generated subgroup G of the group of affine transformations acting on the complex line C, we are interested in the quotient Fix( G)/G. The purpose of this note is to establish when this quotient is finite and in this case its cardinality. We give an application to the qualitative study of polynomial planar vector fields at a neighborhood of a nilpotent singular point.

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