Some results on one-relator groups

REMARK. An element g of a group is divisible by an integer n, or has a divisor ny if g has an wth root in the group. By the length of the relator is meant the letter length of the relator as a word in a free group. R. C. Lyndon [7] has shown that the cohomological dimension of a torsion-free one-relator group is ^ 2 . Now the cohomological dimension of a free abelian group of rank n is n, and of a direct product of an infinite cyclic group with a noncyclic locally cyclic group is >2, (see [6], [2]). Since the cohomological dimension of a subgroup is less than or equal to the cohomological dimension of the group, it follows that the only abelian subgroups of a torsion-free one-relator group are free abelian of rank g 2 or locally cyclic subgroups in which every nontrivial element is divisible by at most finitely many primes. The proof of Theorem 1 uses the usual argument of the Freiheitssatz (see [9]) together with the following ideas. DEFINITION. Let H be a subgroup of G and p a prime. Then H is