I express my thanks to Dr B H Neumann for his generous help, guidance and encouragement, and my gratitude and appreciation to my Parents to whom this opportunity for further study is due.

I take this opportunity to express my gratitude and appreciation to my Parents, without whose help and encouragement this opportunity for further study would have been both unwanted and impossible.

It is a very great pleasure for me to acknowledge the help of Dr B H Neumann during the past 30 months. He suggested this topic to me and his ever-ready advice, criticism and creative remarks have helped, in no small way to make this dissertation possible. It has been both a privilege and a delight to have him as a supervisor. I thank the members of the Department of Mathematics of the University of Manchester for their interest and many useful comments. Finally, I thank the University of the Witwatersrand for a grant in my third year.

A large number of Russian mathematicians have investigated groups satisfying conditions as to the existence and uniqueness of roots in groups. This thesis is concerned with groups in which certain classes of roots are unique. The investigation makes use of notions which belong more to abstract algebra than to group theory. In particular "free" systems of groups with unique roots are studied.

This is a slightly polished version of notes prepared as an aid to a series of lectures on finitely generated nilpotent groups given in May 1969 at the University of Texas, Austin. The notes begin with a short Chapter 0, Basic notions and results. Chapter 1, Algorithmic problems for finitely generated nilpotent groups, is devoted to the word, conjugacy and isomorphism problems for finitely generated nilpotent groups. Chapter 2, Residual properties and some applications, is concerned mainly with residual properties of finitely generated torsion-free nilpotent groups; the existence and uniqueness of the Malcev completion is established. Chapter 3, Lie and associative ring techniques and the commutator calculus, provides a quick account of basic material for use in Chapter 4, Lie group techniques, which is the heart of the notes ...
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The notes convey the spirit of the subject without always going into all the details and the careful reader will often have a good deal of work to do.

... a person of outstanding merit and accomplishment in his field.

In this series of 4 lectures I will not have the time to survey the entire field of modern day one-relator group theory. Instead I have chosen to focus on a few of the facets of the theory that either arise from other areas of mathematics or seem to have wider application than to one-relator groups alone. Much of what I say will involve the prevalence of free subgroups and the way in which properties of free groups persist in one-relator groups. It is clear to me that I will have left many important areas untouched and that often too little time will be devoted to work that deserves more. Indeed there are many important papers that will not even be referred to here and the bibliography is by no means a comprehensive one.

The paper is very well written and is a pleasure to read.

Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. With surprising frequency, problems in a wide variety of disciplines, including differential equations, automorphic functions and geometry, have been distilled into explicit questions about groups, typically of the following kind: Are the groups in a given class finite (e.g., the Burnside problem)? Finitely generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether they are isomorphic or not? The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions. In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. These notes, bridging the very beginning of the theory to new results and developments, are devoted to a number of topics in combinatorial group theory and serve as an introduction to the subject on the graduate level.

The author exhibits some known old and new fascinating theorems in an "elementary" exposition and in almost all cases with complete proofs preceded by the necessary background material. An indication of the material covered is given by the titles of the chapters: Chapter I. History. Chapter II. The weak Burnside problem. Chapter III. Free groups, the calculus of presentations and the method of Reidemeister and Schreier. Chapter IV. Recursively presented groups, word problems and some applications of the Reidemeister-Schreier method. Chapter V. Affine algebraic sets and the representation theory of finitely generated groups. Chapter VI. Generalized free products and HNN-extensions. Chapter VII. Groups acting on trees. The reviewer feels this book will be very useful also to the graduate student and pleasant to read for the mature and expert mathematician.

The emphasis of the seminar is on Combinatorial, Geometric and Computational Group Theory. Many of the talks touch also on three-dimensional topology, homological algebra and theoretical computer science. There are usually 10 seminars each semester. A sprinkling of these seminars are expository and are sometimes combined with one-day or two-day long workshops or conferences.

Groups are mathematical structures, which capture, in mathematical form, the nature of symmetry. It turns out, for this very reason, that they play an important role in crystallography, in particle physics, in geometry and in the study of the three-dimensional world that we live in. They are an important tool in many diverse mathematical disciplines as well as in theoretical computer science. Recently they have been used in cryptography, a field of enormous economic and strategic importance.

Founded by Gilbert Baumslag, its original scope was research at the intersection of Group Theory and Theoretical Computer Science. Following some exciting work applying group theory to secure encryption, the Center's focus shifted to becoming a leading research institution in Cryptography and Network Security. CAISS members do research and publish academic papers in a variety of areas, from the theoretical foundations of cryptography to the design and implementation of cryptographic protocols.

Sadly, our co-author and friend Gilbert Baumslag died during the preparation of this paper. The remaining two authors dedicate the paper to the memory of Gilbert and to all the work that he inspired. In 1985, at Groups St Andrews, Gilbert Baumslag gave a short course on one-relator groups which provided a look at the subject up to that point. In this paper we partially update the massive amount of work done over the past three decades. For the most part we concentrate on areas and results to which the authors have made contributions. We look at the important connections with surface groups and elementary theory, and describe the surface group conjecture and the Gromov conjecture on surface subgroups. We look at the solution by Wise of Baumslag's residual finiteness conjecture and discuss a new Baumslag conjecture on virtually free-by-cyclic groups. We examine various amalgam decompositions of one-relator groups and the Baumslag-Shalen conjectures. We then look at a series of open problems in one-relator group theory and their status. Finally we introduce a concept called plainarity based on the Magnus breakdown of a one-relator group which might provide a systematic approach to the solution of problems in one-relator groups.

While this book was being completed, tragically our colleague, co-author and friend, Gilbert Baumslag, died. Gilbert was one of the foremost researchers in the world in infinite group theory and developed many computational techniques for finitely presented groups. Over the past decade he had been active in transferring this knowledge to group-based cryptography. It is our hope that this book honours his memory and his contribution.

Cryptography has become essential as bank transactions, credit card information, contracts, and sensitive medical information are sent through insecure channels. This book is concerned with the mathematical, especially algebraic, aspects of cryptography. It grew out of many courses presented by the authors over the past twenty years at various universities and covers a wide range of topics in mathematical cryptography. It is primarily geared towards graduate students and advanced undergraduates in mathematics and computer science, but may also be of interest to researchers in the area. Besides the classical methods of symmetric and private key encryption, the book treats the mathematics of cryptographic protocols and several unique topics such as Group-Based Cryptography, Gröbner Basis Methods in Cryptography, and Lattice-Based Cryptography.

He had a great fondness for Nantucket where he spent more than 25 years during school vacations and holidays writing papers, exploring problems on his famous 500 lb blackboard installed on the office wall of his beloved old house on Darling Street. He was enormously fond of many friends on Nantucket with whom he shared a lot of important and silly experiences. One such was playing cricket on Darling Street on a hot August night with four amateurs at the game. There were no winners. There were also no losers. He also loved having lunch at The Opera House where there was no rush, no impatience, and just great food, a great atmosphere, all under the guidance of the renowned Gwen Gaillard. And the always present lure of the Atlantic Ocean was not to be ignored. He loved the land of his birth and was truly happy when the government was transformed by the release of Nelson Mandela. He went to Johannesburg whenever possible for more than 40 years not only to see old friends but also to encourage math projects in the country.

I am deeply shocked by the brutally sudden and unexpected death of Gilbert. He had a long and successful life, being a man of many talents. With his cheerful helpful personality he brought zest to even the most subdued gatherings, bringing them fully to life. He had a major influence on the theory of infinite groups and computing science and produced an abundance of books, articles and research students. His was a unique lecturing style which brought a focus and clarity to the subject. He was good at sport and retained his interest in particular to cricket. He disliked discrimination and as a student in Apartheid South Africa he protected an Indian friend from harassment. Much later on, as Apartheid became even more vicious, he helped get the same man out of a South African prison. As his younger brother he was my hero. I am still in awe of him.