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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...
If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism . A right inverse (section) of is called a coaction on G. In this paper we study and the sections of . We consider the following topics: the structure of as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and the resulting...
The fixed point submonoid of an endomorphism of a free product of a free monoid and cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’s result on the computability of the fixed point subgroup of a free group automorphism is generalized to endomorphisms of free products of a free monoid and a free group which are automorphisms of the maximal subgroup.
The fixed point submonoid of an endomorphism of a free product of a free monoid and
cyclic groups is proved to be rational using automata-theoretic techniques. Maslakova’s
result on the computability of the fixed point subgroup of a free group automorphism is
generalized to endomorphisms of free products of a free monoid and a free group which are
automorphisms of the maximal subgroup.
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