Equations over groups, and groups with one defining relation
This paper contains detailed proofs of important results in combinatorial group theory, at least some of which the author originally obtained as far back as 1979. The first result generalizes Magnus' Freiheitssatz: If A and B are locally indicable groups (i.e. with every nontrivial finitely generated subgroup having an infinite cyclic quotient) and w is any cyclically reduced word in A*B of length \(\geq 2\), then the normal closure of w in A*B intersects A and B trivially. The proof is purely algebraic, along the general lines of Magnus' proof. (A different proof was given independently by \textit{J. Howie} [J. Reine Angew. Math. 324, 165-174 (1981; Zbl 0447.20032)].) The author goes on to construct a nontrivial class \({\mathcal A}\) of locally indicable groups with the property that for each group \(A\in {\mathcal A}\) and each equation over A, there is a group K in \({\mathcal A}\), containing A as a subgroup, in which the equation has a solution. (From this one readily deduces in particular an affirmative answer to the question of Bokut' as to whether there exists a nontrivial group over which every equation has a solution (in the group).) The class \({\mathcal A}\) is quite wide, containing in particular all free groups, and closed under the formation of ''one-relator products'' \(A_ wB\) (the quotient of A*B by the normal closure of w) where A,B\(\in {\mathcal A}\), and w is as above with the additional property that it is not a proper power. It follows that every torsion-free one-relator group is locally indicable (whence in particular appropriate group rings of such groups have no zero-divisors). It is also shown that a group is locally indicable precisely if it contains a subnormal system with torsion-free abelian factors, so that torsion-free one-relator groups have this structure. An example of a one-relator group with a soluble subnormal system (i.e. an ''SN-group''), but without a soluble normal system (i.e. not an ''SI-group''), is given. Several other results of interest are proved.
subnormal system with torsion-free abelian factors
scientific article; zbMATH DE number 3927189