James Howie

On Locally Indicable Groups

A group is said to be locally indicable if each of its non-trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image. Such groups were studied by Higman [-9] in connection with the zero-divisor and unit problems for group rings. More recently, they have arisen I-2, 12] in the study of equations over groups. In [12], I proved a Freiheitsatz for locally indicable groups. This has been proved independently by Brodskii [2] and Short [22]. The present paper arises from an investigation of a question put to me by S.J. Pride whether torsionfree 1-relator groups are locally indicable. The question was raised originally by Baumslag (I-1], Problem 19) and an affirmative solution has recently been announced by Brodski~ [2]. As a consequence, the group algebra RG of a torsion-free 1-relator group G over an integral domain R has no non-trivial zero-divisors, and no non-trivial units (using Higmans results [9]). The first fact was also proved by Lewin and Lewin [16], by embedding RG in a division ring. The second appears to be new, as was pointed out to me by K.A. Brown. A second consequence is that no 1-relator group has a non-trivial finitely generated perfect subgroup, which answers [1], Problem 7 and [10], Question 1. In fact, using the Freiheitsatz, and the tower method described in [12], it is possible to prove the following general version of Brodskiis theorem.

On the asphericity of ribbon disc complements

The complement of a ribbon n-disc in the (71 + 2)-ball has a 2-dimensional spine which shares some of the combinatorial properties of classical knot complement spines. It is an open question whether such 2-complexes are always aspherical. To any ribbon disc we associate a labelled oriented tree, from which the homotopy type of the complement can be recovered, and we prove asphericity in certain special cases described by conditions on this tree. Our main result is that the complement is aspherical whenever the associated tree has diameter at most 3.

One-relator quotients and free products of cyclics

It is proven that the Freiheitssatz holds for all one-relator products of cyclic groups if the relator is cyclically reduced and a proper power. The method of proof involves representing such groups in PSL2(C) and is a refinement of a technique of Baumslag, Morgan and Shalen. The technique allows the extension of the Freiheitssatz result to many additional one-relator products.

The Subgroups of Direct Products of Surface Groups

A subgroup of a product of n surface groups is of type FPn if and only if it contains a subgroup of finite index that is itself a product of (at most n) surface groups.

The genus problem for one-relator products of locally indicable groups

A one-relator product of groups A and B is a natural generalisation of a onerelator group, and under suitable conditions (for example if A and B are locally indicable) one can extend much of the theory of one-relator groups to this more general situation (see for example [3, 15, 16]). The present paper began as an attempt to generalise B.B. Newmans result [21] that the conjugacy problem is solvable for a one-relator group with torsion, but it turns out that our methods yield a much stronger result. Following Culler [8], we define the genus in a group G of an n-tuple (w~ . . . . , w,) of elements of G, written genusG(w I . . . . . w,), to be the least integer g > 0 for which the equation

CONJUGACY OF FINITE SUBSETS IN HYPERBOLIC GROUPS

There is a quadratic-time algorithm that determines conjugacy between finite subsets in any torsion-free hyperbolic group. Moreover, in any k-generator, δ-hyperbolic group Γ, if two finite subsets A and B are conjugate, then x-1 Ax = B for some x ∈ Γ with ǁxǁ less than a linear function of max{ǁγǁ : γ ∈ A ∪ B}. (The coefficients of this linear function depend only on k and δ.) These results have implications for group-based cryptography and the geometry of homotopies in negatively curved spaces. In an appendix, we give examples of finitely presented groups in which the conjugacy problem for elements is soluble but the conjugacy problem for finite lists is not.

The solution of length four equations over groups

Let G be a group, F the free group generated by t and let r(t) E G * F . The equation r(t) = 1 is said to have a solution over G if it has a solution in some group that contains G. This is equivalent to saying that the natural map G -(G * Flr(t)) is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of t in r(t) is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of t in r(t) is equal to four.

A proof of the Scott–Wiegold conjecture on free products of cyclic groups

Abstract Problem 5.53 of Mazurov and Khukhro (Unsolved Problems in Group Theory: The Kourovka Notebook, 12th Edition, Russian Academy of Sciences, Novosibirsk, 1992) (contributed by Wiegold, attributed to Scott) asks whether a free product of three (finite) cyclic groups can be normally generated by a single element. We give a proof of the conjectured negative answer, and an application to Dehn surgery on knots: if Dehn surgery on a knot is S3 gives a connected sum, then all but at most 2 of the connected summands are Z -homology spheres, and hence (by a result of Valdez and Sayari) the number of connected summands is at most 3.

On the Asphericity of Length Five Relative Group Presentations

We investigate asphericity of the relative group presentation 〈G, t | atbtctdtet = 1〉 and prove it aspherical provided the subgroup of G generated by {ab−1, bc−1, cd−1, de−1} is neither finite cyclic nor a finite triangle group. We also prove a similar result for the closely related relative group presentation 〈G, s, t | αsβsγt = 1 = δtεtζs−1〉.

Homological and topological properties of locally indicable groups

The classes of locally indicable groups, conservative groups andD-groups have each been defined in a different context, and have been studied for various reasons. These three classes are shown to coincide. The corresponding mod p versions of the classes are also shown to coincide, for any prime p. Applications to topology are given. In particular, new light is shed on work of Adams on a problem of Whitehead concerning asphericity in 2-complexes.