On torsion-free nilpotent loops

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

Residual nilpotence and ordering in one-relator groups and knot groups

Let G = 〈x, t | w〉 be a one-relator group, where w is a word in x, t. If w is a product of conjugates of x then, associated with w, there is a polynomial Aw(X) over the integers, which in the case when G is a knot group, is the Alexander polynomial of the knot. We prove, subject to certain restrictions on w, that if all roots of Aw(X) are real and positive then G is bi-orderable, and that if G is bi-orderable then at least one root is real and positive. This sheds light on the bi-orderability of certain knot groups and on a question of Clay and Rolfsen. One of the results relies on an extension of work of G. Baumslag on adjunction of roots to groups, and this may have independent interest.

FAITHFULNESS OF CERTAIN MODULES AND RESIDUAL NILPOTENCE OF GROUPS

Let F be a non-cyclic free group and R, S its normal subgroups. We study the abelian group [Formula: see text], viewed as a module over F/RS, via conjugation in F, and residual nilpotence of the group F/[R, S]. An application to the asphericity of finite presentations is given.