Remembrances of Gilbert Baumslag

The material below is extracted from 'Remembrances of Gilbert', ResearchGate DOI: 10.1515/9783110638387-203 (October 2021).

1. Preface

Gilbert Baumslag was one of the leading infinite group theorists of the second half of the twentieth century. Through his own work, his students, and his mentoring, he had a profound effect on both the direction and interests in group theory. There is an ongoing seminar on his work given at Cornell University and various talks from this seminar can be found on the internet. The year he organised on combinatorial group theory at MSRI in 1989 ushered in much of the interest on automatic groups and hyperbolic groups. We note the paper by Alonso, Gersten, Shapiro, and Short explaining Gromov's ideas on hyperbolic groups and which came out of that conference as one of foundational works in geometric group theory. Gilbert's work and influence has been extremely wide, from classical combinatorial group theory, involving embedding theorems, and theory of nilpotent and one-relator groups to automatic and hyperbolic groups; the development of algebraic geometry over groups with Myasnikov and Remeslennikov, which was instrumental in the proof of the Tarski theorems; the use of the variety of group representations with Peter Shalen; and the development of group-based cryptography. In addition to all this work in theoretical group theory, there was also his work on the Magnus project and all that it entailed in terms of computer implementations of group theoretic procedures. Since group theoretic procedures do not necessarily terminate, the Magnus project work as well as the work on group-based cryptography, veered greatly into computer science applications.

2. Ben Fine and Gerhard Rosenberger

Ben Fine began working with Gilbert through the Group Theory seminar as soon as I completed my own thesis work under Wilhelm Magnus. Gilbert was extremely helpful and always willing to listen to work and to give ideas. After the Friday seminar talk and either before or during dinner, a great many ideas for theorems and proofs flowed freely - most coming from Gilbert. He was fiercely protective of his own work as any great mathematician should be, but he was an extremely generous colleague who was willing to help anyone who needed it. Up until his death, he had an active group working with him consisting of both his own students and others. Gerhard Rosenberger and Ben Fine began working with Gilbert in the mid 1980s when some of his work on the variety of group representations with Peter Shalen and our work with Jim Howie on essential representations overlapped. While Ben worked in the CAISS lab, lunchtime conversations with Gilbert led to a great deal more work particularly in group-based cryptography which got to Gerhard via email. We especially like material we did on provable password security. He became our co-author, along with Martin Kreuzer, on our book "A Course in Mathematical Cryptography," published by DeGruyter in 2015.

In 2000, I (Ben Fine) mentioned to Gilbert that the Magnus framework would be ideal with statistical evaluations, where the problems of whether the algorithms terminated or not did not appear. In Gilbert's inimitable style, he said "great idea. You go ahead and do it." That began a 15-year project at CAISS developing CAISS-Stat that involved the programming work of Xiaowei Xu, Yeagor Brjukhov, P C Wong, and others as well as the work of Gilbert and myself. We developed what I felt was an excellent package which if further developed properly would have rivalled SAS and SPSS. What we developed up to a point was excellent, but we learned just how difficult it is to remove bugs from a multiuser software package. We also learned how difficult it is to actually market a new idea - no matter how good it is. I also learned how enthusiastic Gilbert could become. We ran several very nice and instructive seminars on CAISS-Stat. Through all of this work, it was more that just mathematics. We had conversations about all conceivable topics (we agreed on all things political). I considered Gilbert my good friend and mentor and I miss him.

3. Chuck Miller

Looking us up individually on MathSciNet, one finds a graphic showing that we were each others most frequent collaborator. We each had a number of other collaborators, and Gilbert was a lot more productive than I was. But we had a productive and enjoyable collaboration over many years. Our collaboration began with a mistaken example. In about 1974, I was a junior faculty member at Princeton, and Gilbert had returned to CUNY the year before. We saw each other fairly often in the New York Group Theory Seminar. One day he phoned me to tell me about a group that he and Frank Cannonito had constructed because he knew I would be interested.

Their group was (supposedly) locally free but could not be imbedded in a finitely presented group. He described the construction, but I could not immediately see how it worked. After a day or so thinking about this, I came up with a proof that every locally free group could be imbedded in a finitely presented group. So their purported example could not exist. It was also clear the method applied in more general circumstances where all the isomorphisms between finitely generated subgroups can be recursively enumerated. One simply realises all of these isomorphisms in a large HNN extension which is then embeddable in a finitely presented group. After a few phone calls and mail messages, Gilbert was convinced. At his suggestion, we joined forces and eventually published four papers on this and related topics. Years earlier, in 1963-1964, I was a masters degree student in a subject on finitely presented groups that Gilbert taught at NYU.

There were several appealing aspects of Gilbert's subject. Some tools used were free groups, amalgamated free products, and HNN extensions. Basic questions were: when are groups finitely presented or finitely generated and the same for subgroups? One learned these constructions can be used to make new groups with interesting properties; for instance, the non-hopfian Baumslag-Solitar group BS(2, 3) they had recently concocted. Also one studied decision problems and showed the Adian-Rabin theorem that most properties are not recognisable. I found the connections with logic and the ability to construct things of particular interest. After that year, I moved to Illinois which had strengths in algebra and logic, and eventually Bill Boone was my PhD supervisor.

So Gilbert and that one subject had a strong influence on my mathematical interests and career. In 1976, I moved to Melbourne, but managed to visit the US fairly regularly. In the fall of 1979, I was on sabbatical resident at IAS when we began another collaborative effort on the homology of finitely presented groups with Eldon Dyer. We showed that any suitably given sequence of abelian groups could be realised as the homology sequence of a finitely presented group. Although this is a remarkable result, it is frustratingly short of a characterisation. Like all the collaborations I have been involved in, this had elements of social enjoyment as well as mathematical pleasure.

Over the next 10 years, I was heavily involved in administration in Melbourne and visits to New York were less frequent. When I did come to New York, Gilbert often had a project in mind that he hoped we might do something toward. Resulting from one such project was our short paper "Some odd finitely presented groups" which we later jokingly called our most underrated paper. In it, we constructed a nontrivial finitely presented group which maps onto its direct square. So it is non-hopfian. A related group has a finitely generated derived group which is isomorphic to its direct square. Starting in 1991, Gilbert became quite involved in the project to develop the Magnus computational group theory package. During much of that year, I was in New York and was also involved. Among other things, I wrote a collection of programs for manipulating words and subgroups in free groups. I also began collaborations with Gilbert and Hamish Short and eventually Martin Bridson.

After 2000, my obligations in Melbourne lessened and I visited New York more regularly. Gilbert was still concerned with the Magnus project and funding, but he managed time for lots of small research projects. He had a talent for suggesting projects one could make progress on in finite time. The results were generally interesting if not all that deep. He was also very generous with help for students and colleagues. For one such project, we were trying to better understand a group constructed by B H Neumann in the 1930s. But we only talked about this group while riding on the crosstown bus. We could just pick up where we had left off on a previous trip. This was good sport. With all the mathematical jargon, we were sure other passengers thought we were very strange.

Gilbert was a good friend and a very talented mathematician. We enjoyed each others company and doing mathematics together. He was remarkably generous with ideas and particularly helpful to students.

There is another aspect of his personality that I admired. When dealing with someone at say a counter in an office or a teller or an official, he would chat with them in an understanding way that indicated appreciation of their circumstance. It often put them at ease and made their lives seem a bit brighter, as though he was spreading a little joy.

4. Martin Bridson

Gilbert Baumslag was my friend. We shared passions - for cricket, for arguing about how the world should be, for a measure of carousing, and above all for mathematics, especially the mathematics of group theory, about which he taught me a very great deal. For many years, starting in 1989, it was through conversation that he nurtured my appreciation of the elegance and achievements of pre-geometric group theory. But his great influence on our subject is in large part a function of his ability to express his many insights with such clarity in his writing. He was a master of explaining his solution to hard problems by exhibiting concise examples, and he had a well-honed skill for illuminating key ideas without stinting on rigour.

It is remarkable to reflect on how often Gilbert's papers opened doors to new worlds that held many wonders. One such paper that has been of particular significance in my life is Gilbert's article with Jim Roseblade on the subgroups of a direct product of two non-abelian free groups. Gilbert and Jim showed that, on the one hand, the array of finitely generated subgroups is strikingly diverse, while on the other hand, the only finitely presented subgroups are the obvious ones. This insight, and the homological techniques used to establish it, were the first key steps in what became a remarkably rich theory of subdirect products of hyperbolic and related groups, a subject that was the scene of many adventures for Gilbert, myself, Jim Howie, Chuck Miller, and Hamish Short (in various combinations) over 15 years.

Gilbert and I built many groups together but only published four papers. This is an inadequate indication of how much his mathematics influenced mine. A more accurate picture emerges if one scans the list of references in my other papers: he is there in many of them, for a variety of reasons - sometimes for his work on isoperimetric inequalities, subdirect products or equations over groups, but also for his seminal work on metabelian and nilpotent groups, profinite completions, and non-hopfian groups. His insights have populated my universe with great characters that I would not otherwise have known. My son, James, is a gifted cricketer in whom Gilbert took a particular delight. After Gilbert died, I pinned an email from Gilbert next to a picture on my office wall showing James in the act of bowling. I had shared that picture with Gilbert and his typically brief reply was one of the last things he wrote to me: "I envy you James. Love to you both, Gilbert."

"We are all completely mad," he would often say. But oh what a life-affirming madness it was. I miss him greatly.

5. Anthony Gaglione and Dennis Spellman

Dennis recalls attending a talk of Paul Schupp in which he was discussing groups with a certain property - call it P. (Dennis does not remember what P was!) Schupp's theorem was that finitely presented (or maybe finitely generated?) groups are generically - P. He justified the importance of his theorem with the assertion that unless you are Gilbert Baumslag, concrete examples of such groups are notoriously difficult to find. Such was the respect for Gilbert's power among his contemporaries. Not only does Gilbert's prolific outpouring of significant research merit recognition but also his encouragement of others. Dennis also recalls the late Seymour Lipschutz relating to him Seymour's discussing a result of his with Gilbert and Gilbert replying that the result implies a solution to the conjugacy problem for cyclically-pinched one relator groups and that Seymour must publish. Thus, also in his encouragement of others, Gilbert helped advance the frontiers of our science. In the New York Group Theory Seminar, Magnus had posed the question of whether or not the surface groups are residually finite. In a paper published in 1962, Gilbert established the stronger result that the surface groups are residually free. Here, he introduced the technique of big powers. For this, we are forever in his debt as we use that very technique over and over again in our own work. Implicit in Gilbert's 1962 paper is the fact that the surface groups are not only residually free but even freely discriminated in the sense that one can distinguish finitely many distinct elements by mapping into a free group. This paper along with a paper published in 1967 by his brother Benjamin was the basis for our result (independently proven by Remeslennikov) that, among finitely presented non-abelian groups, the freely discriminated ones are precisely the universally free examples. Incidentally, having mentioned the New York Group Theory Seminar, we would be remiss in not pointing out that that seminar's storied longevity is in no small measure due to Gilbert's stewardship for many years. Consulting Hanna Neumann's classic monograph on "Varieties of Groups," we see that Gilbert along with Bernhard, Hanna and Peter M Neumann introduced (what we shall now call) varietally discriminating groups. ...

6. Doug Troeger

CAISS was founded to bring together the Science and Engineering communities at City College for the sharing of gains in mathematics, computer science, and related disciplines, and exploring the practical extensions of these gains. Gilbert Baumslag was the founding director of CAISS.

Building on the software package MAGNUS developed under Baumslag's leadership in the 1990s, the primary goals were to apply algebra and group theory to a wide array of applications, and to extend and expand the approach to computer algebra embodied in MAGNUS.

CAISS' ties to computer science were formalised with Baumslag's transfer to the Department of Computer Science in 2007. He was the key to recruiting N Fazio, R Gennaro, L Gurvits, and W Skeith to join the computer science faculty, with simultaneous appointment to CAISS. CAISS remains a vital Center at City College; R Gennaro currently serves as director. ...

CAISS contributed to the bridging of the mathematics and computer science departments as well by organising a number of conferences and workshops of interest to both and held on the City College campus. These included Software for the Working Mathematician [2003], Axiom Conference [2005], Computation and Complexity [2006], Visualization Day [2008], Security and Privacy Day [2011], and Faces of Modern Cryptography [2011]. In addition, CAISS maintained a small Linux laboratory for the use of undergraduates, facilitating their participation in many of the Center's projects. T Daly and R Bruykhov offered courses on open-source programming using this lab, which was also used to host summer research programs for talented high school students.

Last Updated February 2023