Abstract

The notion of reduced diagram plays a fundamental role in small cancellation theory and in tests for detecting the asphericity of 2-complexes. By introducing vertex reduced as a stricter form of reducedness in diagrams we obtain a new combinatorial notion of asphericity for 2-complexes, called vertex asphericity, which generalizes diagrammatic reducibility and implies diagrammatic asphericity. This leads to a generalization and simplification in applying the weight test [2] and the cycle test [6] [7] to detect asphericity of 2-complexes and (for the hyperbolic versions of these tests) to detect hyperbolic group presentations. In the end, we present an application to labeled oriented graphs. We would like to thank the referee for his helpful suggestions.

Basic Definitions

A p.l. map between 2-complexes is called combinatorial, if each open cell is mapped homeomorphically onto its image. A 2-dimensional finite CW-complex is called combinatorial, if the attaching maps of the 2-cells are combinatorial relative to a suitable polygonal subdivision of their boundary.

Let KP be the standard 2-complex of the presentation P (we assume all presentations to be finite). A diagram is a combinatorial map f: M → KP, where M is a combinatorial subcomplex of an orientable 2-manifold. A spherical diagram is a diagram f:S → KP, where S is the 2-sphere. These definitions may be found for example in [1], [2], [6], [7] or [8].

Introduction

Suppose G = 〈x1, …, xn; R = 1〉 is a one-relator group with R a cyclically reduced word in the free group on {x1,…,xn} which involves all the generators. The classical Freiheitssatz or independence theorem of Magnus (see [25]) asserts that the subgroup generated by any proper subset of the generators is free on those generators. More generally suppose X and Y are disjoint sets of generators and suppose that the group A has the presentation A = 〈X; rel(X)〉 and that the group G has the presentation G = 〈X,Y;rel(X),Rel(X,Y)〉. Then we say that G satisfies a Freiheitssatz relative to A if < X >G= A, that is the subgroup of G generated by X is isomorphic to A. In this more general language Magnus' result says that a one-relator group satisfies a Freiheitssatz relative to the free group on any proper subset of generators.

A great deal of work has gone into proving the Freiheitssatz for the class of one-relator products. These are groups of the form G = (A * B)/N(R) where R is a non-trivial,cyclically reduced word in the free product A * B of syllable length at least two. A and B are called the factors and R is called the relator. In this case the Freiheitssatz means that A and B inject into G.

Residually finite varieties of groups were completely described, in [1], by Ol'shanskii. He proved that a group variety is residually finite if and only if it is generated by a finite group with abelian Sylow subgroups.

The next question is: “Which varieties are locally residually finite?” Hall [9] proved that all finitely generated abelian-by-nilpotent groups are residually finite. Hall formulated a conjecture that his result can be extended to the class of abelian-by-poly cyclic groups. Jategaonkar [2] proved that finitely generated abelian-by-polycyclic groups are residually finite.

The following result was obtained by Groves [8]. Let Tp be the variety generated in the variety ℬpA by all 2-generated groups belonging to ZA2, (p an odd prime), and let T2 be the variety generated in the variety A by all 2-generated groups belonging to ZA2A.

Theorem 1(Groves) If W is a variety of metanilpotent groups then the following conditions are equivalent.

1. W does not contain any Tp.

2. W is locally residually finite.

3. All finitely generated groups in W satisfy the maximal condition for normal subgroups.

In [4] it was proved that for odd primes p the variety Tp coincides with ZA2 ∩ ℬpA, and T2 was also described in the language of identities.

Conjecture 1The only minimal, non-locally residually finite, varieties of solvable groups are the varieties from the previous theorem and the varieties ApAqA (p, q are distinct primes).

‘This is a truly wonderful book, which unfortunately our library is too small to contain.’ -Pierre de Frontage

In the spring of 1993, a 10-day workshop was held at Heriot-Watt University, Edinburgh, under the auspices of the International Centre for Mathematical Sciences, on Geometric and Combinatorial Methods in Group Theory. This volume contains papers contributed by participants at the workshop. Some report work presented by the authors in lectures at the conference, and all of them are on topics closely related to the central theme of the conference. Survey articles were kindly contributed by S M Gersten, R I Grigorchuk, P H Kropholler, A Lubotzky, A A Razborov and E Zelmanov, who were among the invited conference speakers.

The problem section at the end of the book is made up of problems presented at a problem session on the final day of the conference. The session was chaired by S J Pride, and the list was compiled using notes taken by S Wreth together with written comments from the presenters of the problems.

The editors are deeply indebted to V Metaftsis and S Wreth for their invaluable assistance in the running of the conference. We could not have succeeded without their help. In addition, we are grateful to the many other people who helped to make the conference a success.

Abstract

An algorithm is described for determining the indivisible Nielsen paths for a train track map and therefore the subgroup of elements of the fundamental group fixed by the induced automorphism.

In my talk at the Edinburgh Conference I described the “procedure” for finding fixed points of an automorphism of a free group that is implicit in [6] and explicit in [2] and [7]. I made the point that this is a procedure and not an algorithm since there is no way in general of knowing how long to persist before being sure that all fixed elements have been found—although it has been shown to be effective for positive automorphisms [2]. The example of Stallings [8], p99, figure 3 (Example 1 below) was presented to show that one may need more persistence than expected. I also discussed the notion of an indivisible Nielsen path (INP) which was introduced in [1] and used as a fundamental tool in [4]. After the talk, Bestvina asked me whether the procedure could be adapted to determine the INPs for a train track map, the determination of which is an essential part of the algorithm introduced in [1] for finding the fixed words of the induced automorphism. This paper shows how to do this. All irreducible automorphisms have train track representatives so this provides a straightforward means (given the train track map!) of computing the generator of the fixed subgroup.

Introduction

The purpose of this paper is first to give a survey of some recent results concerning semigroup presentations, and then to prove a new result which enables us to describe the structure of semigroups defined by certain presentations.

The main theme is to relate the semigroup S defined by a presentation II to the group G defined by II. After mentioning a result of Adjan's giving a sufficient condition for S to embed in G, we consider some cases where S maps surjectively (but not necessarily injectively) onto G. In these examples, we find that S has minimal left and right ideals, and it turns out that this is a sufficient condition for S to map onto G. In this case, the kernel of S (i.e. the unique minimal two-sided ideal of S) is a disjoint union of pairwise isomorphic groups, and we describe a necessary and sufficient condition for these groups to be isomorphic to G.

We then move on and expand on these results by proving a new result (Theorem 9), which is a sort of rewriting theorem, enabling us to determine the presentations of the groups in the kernel in certain cases. We finish off by applying this new result to certain semigroup presentations and by pointing out its limitations.

The Story

If any of you who read this volume do not like stories, then I am sorry for you. Stories are the thread from which the fabric of the world is woven, and to dislike stories is to dislike life. But, to any such people, I would also say that if you read this story you will also learn some mathematics.

Once there was a mathematician who had little wisdom. One spring he attended a group theory conference in Scotland, which may or may not have been wise of him. Since the conference was long, he decided to take a couple of days off, which was certainly wise. He had heard much about the beauty of Scotland's rivers, and the fine salmon that swam in them, so he decided to go salmon-fishing. He did not think of the need for a licence, nor that a large charge is made for the right to fish for salmon in most places; indeed, he had not even checked whether there were salmon in the rivers at that time of year. This may seem foolish of him, but turned out not to be so.

He went to the Tweed, which was running sweetly. He saw many people fishing for salmon, but found a pool where no-one was. Not thinking that this might be because that was not a good place for salmon (for he had little wisdom, though, as we shall see, he was also lucky), he began fishing under a bright spring sky.

Let Mn be an n-dimensional smooth manifold in the (n + 2)-sphere Sn + 2, and let Sn + 2 – N(Mn) be the complement of an open regular neighbourhood of Mn in Sn + 2. A Seifert manifold H of Mn is a compact orientable (n + 1)-dimensional submanifold of Sn + 2 such that the boundary of H is Mn. Moreover, if the induced homomorphism from π1(H) into π1(Sn + 2 – N(Mn)) is one-to-one, then the Seifert manifold H is called minimal.

One important case is when Mn is a sphere, i.e. a knot. The existence of Seifert manifolds of knots of any dimension is known. It is well known that the Seifert surfaces of 1-knots with minimal genus are minimal. Gutierrez [1] asserted the existence of minimal Seifert manifolds for knots of any dimension. However, his proof has a gap. For n ≥ 3, Silver [6] has given examples of n-dimensional knots with no minimal Seifert manifolds. Necessary and sufficient conditions for an n-knot (n ≥ 3) to have a minimal Seifert manifold have been given in [7].

In this paper we prove the following result:

Theorem. For any integer g ≥ 1, there exist closed orientable surfaces of genus g in S4 which have no minimal Seifert manifolds and no trivial 1-handles.

This result was presented to KOOK topology seminar in Osaka in September 1989, and at KNOT 90 in Osaka in 1990. The existence of 2-knots with no minimal Seifert manifolds is still an open question.