Gilbert Baumslag

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30 April 1933
Johannesburg, South Africa
20 October 2014
New York City, New York, USA

Gilbert Baumslag had a long career as one of the leading researchers on infinite group theory. In the last part of his career he used his skill in group theory applying it to cryptography.


Gilbert Baumslag was the son of Boruch Baumshlag or Baumslagk (about 1900-1980) and his wife (born about 1910). Boruch Baumshlag was brought up in Daugavapils in the Russian Empire, a city which became part of Latvia in 1920, while his wife was also brought up in the Baltic States. They emigrated in around 1928 and settled in South Africa. Gilbert was the oldest of his parents' three children, having a younger brother Benjamin Baumslag who became a group theorist working in much the same area as Gilbert, and a younger sister.

Gilbert was born in Johannesburg and attended schools in that city. Graduating from high school in 1950, he entered the University of the Witwatersrand (Wits) to study mathematics. This university, founded as the South African School of Mines in 1896, had been granted full university status in 1922. The University had a policy to treat everyone equally, irrespective of their class, wealth, race or creed. In 1948 the South African government began to introduce apartheid policies which the universities opposed. Ismail Mohamed was a student on the same course as Gilbert and he had taught himself group theory despite it not being part of the course at Wits. He persuaded the university to put on an honours group theory course which Gilbert took. It was his introduction to the subject he would spend his career working on. We explain below how, in 1989, Gilbert supported efforts to free Ismail Mohamed from prison. Baumslag was awarded a bachelor's degree by the University of the Witwatersrand in 1953 and he continued to study for a master's degree which he was awarded in 1955.

Baumslag was accepted by the University of Manchester to study for a Ph.D. and he sailed from Durban, South Africa, on the ship the Stirling Castle, arriving in Southampton, England, on 16 September 1955. He stated on entry to England that he was going to the University of Manchester as a student. At this time Manchester was an exciting place for postgraduate work on group theory. Max Newman held the chair of Mathematics and worked on combinatorial topology, but he had undertaken work on computers during World War II and had appointed Alan Turing to Manchester. He had also appointed to Manchester, among others, B H Neumann who was one of the leading group theorists. When Baumslag arrived in Manchester he was immediately introduced to a fellow Ph.D. student of B H Neumann, namely Jim Wiegold. Ian David Macdonald and Michael Frederick Newman, who would become important figures in the development of group theory, joined them a couple of years later, also as students of B H Neumann. While Baumslag was undertaking research at Manchester, Joachim Neubüser spent a postdoctoral year 1957-58 there. Jim Wiegold, who had undertaken research on similar topics to Baumslag, submitted his doctoral dissertation entitled Nilpotent products of groups with amalgamations in 1958 and was awarded his Ph.D.

Baumslag had two papers in print before the award of his doctorate, namely A theorem on infinite groups (1957) and Finite factors in infinite ascending derived series (1958). The 1958 paper was submitted in 1 November 1956 and contains the following acknowledgement:-
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.
While studying at Manchester, Baumslag returned to South Africa in the summer of 1956, sailing from Southampton to Cape Town on the Carnarvon Castle on 7 June, returning on the ship the Capetown Castle from Cape Town to Southampton arriving on 5 October 1956.

He was awarded his Ph.D. in 1958 for his thesis entitled Some aspects of groups with unique roots. He writes in the thesis:-
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.
The thesis has the following Abstract:-
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.
We note that Baumslag's fellow postgraduate students Ian D Macdonald and Mike Newman graduated with PhDs in 1960; Ian Macdonald with the thesis On Groups with Conditions like Finiteness of Conjugacy Classes and Mike Newman with the thesis On Groups All of Whose Proper Homomorphic Images are Nilpotent.

After the award of his Ph.D., Baumslag spent the year 1958-59 as a lecturer in Manchester before returning to South Africa where he married Sybil on 3 July 1959. She was a musician and a fine pianist. Gilbert and Sybil Baumslag had two children but we note at this point that Gilbert and Sybil were divorced on 17 September 1971 at Harris, Texas, USA. The newly married couple sailed on the Stirling Castle from Cape Town to Southampton, arriving on 11 September 1959. They sailed from Southampton on 12 September on the Mauretania arriving in New York on 18 September. They were travelling to Princeton where Baumslag had been appointed as an Instructor in Mathematics. In 1961 he was appointed as an assistant professor at the Courant Institute of New York University, where his address was 6610 Yellowstone Boulevard, Forrest Hills 75, New York. Baumslag and his wife spent the summer of 1961 in Europe, returning to the United States on 23 August 1961. At this point they considered their time in the United States not to be permanent since they gave their permanent address as 199 Oxford Road, Dunkeld, Johannesburg, South Africa.

After a year, Baumslag was promoted to associate professor. In 1964 he became a professor at the City University of New York (CUNY), spending from September 1968 to May 1969 at the Institute for Advanced Study at Princeton. In May 1969 he gave a series of lectures on finitely generated nilpotent groups at the University of Texas, Austin, which he wrote up as the book Lecture notes on nilpotent groups (1971). Mike Newman wrote in the review [17]:-
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 ...
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.
Then, from 1969 to 1973, he was a professor at Rice University. Returning to New York, he was a Distinguished Professor at the City College of the City University of New York from 1973. He was described by CUNY as:-
... a person of outstanding merit and accomplishment in his field.
Baumslag became an American citizen on 2 July 1980. By this time he had around 85 publications in print, about half of the total number he would produce throughout his career. MathSciNet lists 50 co-authors involved in these 166 publications and over his career he supervised the doctoral studies of over 30 students.

In May 1983 Colin Campbell and I [EFR] invited Baumslag to be one of five main speakers at the conference Groups St Andrews 1985 which we organised. All accepted our invitation and Gilbert chose to give his course of four lectures on A survey of groups with a single defining relation. He explained [3]:-
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.
Stephen Pride, in a review of the paper, wrote:-
The paper is very well written and is a pleasure to read.
Let me [EFR] describe an event related to Groups St Andrews 1985 that involved Gilbert Baumslag. Ismail Mohamed was a coloured South African mathematician who specialised in group theory. Like Gilbert Baumslag, he studied at the University of the Witwatersrand, obtaining a B.Sc. and an M.Sc. He obtained a scholarship to study for a Ph.D. at the University of London which he was awarded in 1960. His thesis was entitled On Series of Subgroups Related to Groups of Automorphisms, and his thesis advisor had been Kurt Hirsch. He was appointed as an assistant lecturer at the University of London but, after the racial riots in South Africa, chose to return to a position at the University of the Witwatersrand, turning down the offer of a permanent position at the University of London. Back in South Africa he campaigned for human rights. Mohamed wrote to us saying he wished to attend Groups St Andrews 1985 and we invited him to lecture to the conference. On 18 February 1985 Mohamed was arrested along with other leaders of the United Democratic Front so could not attend Groups St Andrews 1985. During the conference I was approached by some participants who wanted to write to the South African government arguing for Mohamed's release as a prisoner of conscience and wanted to invite other conference participants to sign their letter. I had a worry about this for, at that time, I had no idea what Gilbert Baumslag's political views were and I had no wish to offend him. Gilbert, however, told me about his attempts to help people who were being discriminated against. He said he had known Mohamed as a student at the University of the Witwatersrand, he fully supported the letter and would add his signature in support of Mohamed. The letter was sent off signed by around 300 of the world's leading group theorists. I doubt if it had any impact but the trial of United Democratic Front members collapsed (some of the State's evidence was suspect) and all were released by June 1986.

In 1989 Baumslag organised a year long conference on combinatorial group theory at the Mathematical Sciences Research Institute in Berkeley, California. This led to much increased interest in automatic groups and hyperbolic groups.

In 1991 Baumslag published the book Topics in combinatorial group theory. As with the earlier book mentioned above, this was the result of a course he had given, this time at ETH Zürich in the winter term of 1987/88. The Publisher, following the Preface, writes:-
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.
Stelios Andreadakis writes in the review [1]:-
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 New York Group Theory Seminar was started by Wilhelm Magnus in around 1955 but after he retired in 1987 Baumslag took over running the seminar. In a grant application he made in 2003 he wrote:-
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.
Later in his career, Baumslag became interested in the interaction between group theory and computer science. He founded the Center for Algorithms and Interactive Scientific Software (CAISS) as one of the research centres of the City College of New York:-
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.
In 1994 Baunslag headed a team developing MAGNUS, a computer algebra system specifically designed to compute with infinite groups. It was released to the public in 1997 but in August 2005 it was decided to abandon the project.

Gilbert Baumslag's work on cryptography led to him publishing papers on the topic, the first three of which all appeared in 2006: Cryptosystems using linear groups; A proposed public key cryptosystem using the modular group; and Designing key transport protocols using combinatorial group theory. We note that by this time he was 73 years old but was able to continue working after passing the usual retirement age. In fact after serving in the Department of Mathematics for more than 35 years, he joined the Department of Computer Science at CUNY in 2006.

Baumslag was married to Mary Kennedy and they lived in Manhattan and Long Island. He continued to undertake research until August 2014 when his illness was diagnosed; he died in October 2014. The tenth in the Groups St Andrews series of conferences was planned to be held in Birmingham in 2017 and Gilbert Baumslag, Ben Fine and Gerhard Rosenberger were working on the paper One-relator groups: an overview at the time of his death [4]:-
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.
Another project Baumslag was working on at the time of his death was the book A course in mathematical cryptography co-authored with Ben Fine, Martin Kreuzer and Gerhard Rosenberger. It was published in 2015 and contains the following by Baumslag's three co-authors [5]:-
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.
The Publisher gives the following description of the book [5]:-
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.
For Baumslag's interests outside mathematics we quote from [7]:-
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.
In [16] there are many tributes paid to Gilbert Baumslag. For example, Benjamin Baumslag, his younger brother, writes:-
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.
For more memories of Gilbert Baumslag, see THIS LINK and for more memories from his students, see THIS LINK.

References (show)

  1. S Andreadakis, Review: Topics in combinatorial group theory, by Gilbert Baumslag, Mathematical Reviews MR1243634 (94j:20034).
  2. G Baumslag, Topics in Combinatorial Group Theory (Springer Science & Business Media, 1993).
  3. G Baumslag, A survey of groups with a single defining relation, in C M Campbell and E F Robertson (eds.), Groups St Andrews 1981 (Cambridge University Press, Cambridge, 1985), 30-58.
  4. G Baumslag, B Fine and G Rosenberger, One-relator groups: An overview, in C M Campell, M R Quick, C W Parker, E F Robertson and C M Roney-Dougal (eds.), Groups St Andrews 2017 in Birmingham (Cambridge University Press, Cambridge
  5. G Baumslag, B Fine, M Kreuzer and G Rosenberger, A course in mathematical cryptography (De Gruyter, Berlin, 2015).
  6. F B Cannonito, Some algorithms for polycyclic groups, in C M Campbell and E F Robertson (eds.), Groups St Andrews 1989 Vol 1 (Cambridge University Press, Cambridge, 1991), 76-83.
  7. Gilbert Baumslag, The City University of New York.
  8. Gilbert Baumslag Obituary, New York Times (22 October 2014).
  9. Gilbert Baumslag, Department of Computer Science, The City College of New York, City University of New York.
  10. Gilbert Baumslag, Institute for Advanced Study.
  11. Gilbert Baumslag,
  12. Gilbert Baumslag, in American Men & Women of Science. A biographical dictionary of today's leaders in physical, biological and related sciences (33rd Edition) (Cengage Learning, Detroit, 2015).
  13. Gilbert Baumslag, in Who's Who in America 2003 (Marquis Who's Who, New Providence, NJ, 2002).
  14. In Memoriam: Gilbert Baumslag, Center for Algorithms and Interactive Scientific Software, The City College of New York, City University of New York.
  15. P Longobardi, Review: A Course in Mathematical Cryptography, by Gilbert Baumslag, Benjamin Fine, Martin Kreuzer and Gerhard Rosenberger, Mathematical Reviews MR3558843.
  16. Memories of Gilbert Baumslag,
  17. M F Newman, Review: Lecture Notes on Nilpotent Groups, by Gilbert Baumslag, Mathematical Reviews MR0283082 (44 #315).
  18. Remembrances of Gilbert, ResearchGate (October 2021).

Additional Resources (show)

Honours (show)

Honours awarded to Gilbert Baumslag

  1. Speaker at Groups St Andrews 1985

Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update February 2023