The classical results of Magnus on one-relator groups (such as the Freiheitssatz and the solvability of the generalized word problem for Magnus subgroups) are generalized to some (nitely presented groups whose relators are constructed inductively. The diagrammatic asphericity of such group presentations is also proven. c © 2001 Elsevier Science B.V. All rights reserved. MSC: Primary 20F05; 20F0...

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let F be free group, and let w∈F nonprimitive element. The primitivity rank w, π(w), is the smallest subgroup containing w as an imprimitive Then any group G=F∕⟨⟨w⟩⟩ generated by fewer than π(w) elements free. In particular, if π(w)>2, then G does not contain Baumslag–Solitar hypothesis that π(w)>2 implies presentation co...

Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residual...

We study the asymptotic behaviour of a finitely presented Rees quotient S = Inv〈A | ci = 0 (i = 1, . . . , k) 〉 of a free inverse semigroup over a finite alphabet A. It is shown that if the semigroup S has polynomial growth then S is monogenic (with zero) or k ≥ 3. The three relator case is fully characterised, yielding a sequence of two-generated three-relator semigroups whose Gelfand-Kirillov...

with the property that for any pair r, s of elements of R either r = s or there is very little free cancellation in forming the product rs. The classical example of such a group is the fundamental group of a closed orientable 2-manifold of genus k. The study of this group has given rise both to the theory of one-relator groups, initiated by Magnus ([3]), and to the theory of small cancellation ...