Our book on “Two-dimensional Homotopy and Combinatorial Group Theory (LMS 197)” appeared 24 years ago. Since the turn of the millennium we were encouraged to publish in a similar style the (remarkable) progress that has been obtained in the meantime. On the other hand, none of the fundamental open questions like the Andrews-Curtis Conjecture and the Whitehead-Asphericity problem have been (fully) solved, and almost no prognosis is possible about when it will finally happen. We hope that mathematicians, who are in danger of leaving the subject because of this perspective, will be kept active by this sequel publication; and new ones will be gained for our important field.

Again, we mix surveys and original material in the chapters, which can be read by students and experts alike. Ongoing research is mentioned as work in progress. There are ample references to the LMS 197 volume which will be cited as [I] including the bibliography and the index, where new publications and new notions are listed.

Since the publication of LMS 197 there have been workshops of the authors at least every fifth year. We met in Germany, Italy, Russia and the USA, and the fruits of these meetings are the basis of this book. During a meeting close to Naumburg (Saale) we made the decision to start what you can read now.

A mathematically energetic member of the team of authors was NancyWaller. But when the actual writing had to start, she unexpectedly died. Her preprint [BeWa13] together with F. Rudolf Beyl could not be finished, and in particular it was impossible without her to write a planned chapter on the geometric realization problem for algebraic 2-complexes at the same time. Hence “Nancy's Toy”, which she hoped that would result in a counterexample, has been left to the future. What can be said at present, is collected in an appendix to chapter I by F. Rudolf Beyl and Jens Harlander. This book can be considered as serving for Nancy Waller's memory.

Abstract. We define small cancellation conditions W* and W representing non-positive curvature in cancellation diagrams. Each of these two conditions, which are dual to each other, generalizes the classical small cancellation conditions C(6), C(4) T(4), C(3) T(6). W also contains the small cancellation condition W(6) of Juhasz. Our main result is the solution of the conjugacy problem for groups with a presentation satisfying W* or W. Its proof uses the geometry of non-positively curved piecewise Euclidean complexes developed by Bridson. Following a sketch of Gersten, we also give a detailed proof of the solvability of the word problem for such groups by a quadratic isoperimetric inequality.

THE CONDITIONS W* AND W

Let P = 〈x1,…, xn |R1, …, Rm〉 be a finite presentation of the group G. We always assume that each relator Ri is cyclically reduced and no relator is the trivial word or a cyclic permutation of another relator or of the inverse of another relator. (By “relator” we always mean a denning relator of the presentation.) Let F be the free group on the generators. Kp denotes the standard 2–complex modeled on P.

The Whitehead graph Wp of Kp is the boundary of a regular neighborhood of the only vertex of Kp. For each generator xi of P it has two vertices +xi and –xi which correspond to the beginning and the end of the oriented loop labeled Xi in Kp.

In this chapter, classical decision problems such as the word and conjugacy problem are introduced and methods are given for solving them in certain cases. All the methods we present involve Van-Kampen diagrams as one of the most powerful tools when dealing with the classical decision problems.

Introduction

In 1912, Max Dehn formulated in his article “Über unendliche diskontinuierliche Gruppen” (“On infinite discontinuous groups”) three fundamental problems for infinite groups given by finite presentations: the identity problem, the transformation problem, and the isomorphism problem. The following is a translation of Dehn's definition of the first two problems called in modern terms the word problem and the conjugacy problem:

1. The identity problem (word problem): Let an arbitrary element of the group be given as a product of the generators. Find a method to decide in a finite number of steps whether or not this element equals the identity element.

2. The transformation problem (conjugacy problem): Any two elements S and T of the group are given as a product of the generators. Find a method to decide whether or not S and T are conjugate, i.e., whether or not there exists an element U of the group that satisfies the equation S = UTU-1.

In this chapter we wish to give an account of some recent developments that use Dehn's original geometric ideas in extending partial solutions of the word and conjugacy problems, in particular some recent generalizations of smallcancellation theory.