# The Gilbert Group

**Definition**

*The Baumslag Group*: advisees and protegees of Gilbert Baumslag during his last few decades; Kati Bencsath, Mariana Bonanome, Anthony Clement, Peggy Dean, Sal Liriano, Steve Majewicz, Gretchen Ostheimer, Marcos Zyman.

**1. Beginnings**

*Peggy*: I first met Gilbert when I was an undergraduate and he was my Abstract Algebra professor in 1975. What joy that class was! Gilbert led us into wonderland and let us explore. I also took non-Euclidean geometry with Gilbert, and again he guided us through strange and marvellous places. Gilbert became my unofficial advisor at that time, and it was he who explained that I must go to graduate school, and helped me make it happen.

*Gretchen*: Gilbert was a mentor to me. My advisor took me to meet him in 1993 to get feedback about a problem we had identified as a possible thesis. Gilbert said it was a hard problem, maybe too hard. And then, when I solved it and started writing it up, I discovered that Gilbert himself had solved it many years prior! We joked that that's one liability of having such a huge body of work it was impossible for him to remember even his own results! It all worked out well: I was able to generalise the result and to begin my career as an independent researcher.

*Steve:*Every Friday, the Graduate Center offers various seminars. One of them was the Group Theory Seminar. I think it was in 1998 that I attended my first seminar. I remember being one of several students who would join the professors for coffee and cookies before the seminar (you can guess who some of the other students were - the Baumslag Group). I was not very sociable at the time, so I didn't interact with anyone. One person in the crowd with an accent that I was unfamiliar with attracted a lot of attention, including my own Gilbert. He was very comical and intelligent, and seemed to be the kind of person that I would like to learn from. I did not know if he taught at the GC, nor did I know his teaching style. Luckily for me, he was the speaker at the seminar. He gave a fascinating and intriguing talk in the area of combinatorial group theory. I immediately became interested in learning more about the subject, and I wanted Gilbert to teach me.

*Anthony:*I met Gilbert Baumslag between 2000 and 2001, when I took a group theory course he was teaching. I was working toward becoming an algebraic topologist. After my first class with Gilbert, I immediately changed my mind. Gilbert's teaching was smooth and captivating. He made group theory not only accessible, but even somewhat easy for me to follow. His style of joking throughout his lectures succeeded in making his audience feel completely relaxed and engaged as we learned.

*Marianna:*In 2001, I took a combinatorial group theory course at the CUNY Graduate Center and was fortunate to have Gilbert as my professor. What a magical time! I was finding my way as a graduate student and walking into Gilbert's class was like coming home.

*Marcos:*I am forever honoured and humbled to have crossed paths with Gilbert Baumslag. In 2001, I audited a course he gave on combinatorial group theory at the CUNY Graduate Center. I had never seen such remarkable teaching in my life. It was because of him that I became a PhD student. Ever since that initial encounter, Gilbert's teaching and advice marked me in a profound way, both mathematically and personally. I never stopped attending his lectures from then on.

**2. Teaching**

*Marianna:*Gilbert was dynamic, supportive and full of humour in the classroom. He wrangled his students into group theoretic explorations requiring patience, skill, and creativity. He made every topic new at each revisiting and we his students happily revisited with him. He made me understand that it is okay to pull apart an idea, a theorem, a proof, a construction, again and again for weeks and years at a time. For years, I continued to take every class I could with Gilbert, even after graduating, sometimes taking the same class several times. I still return to my notes from these classes. They are prized possessions. Gilbert's spirit lives on in all of our teaching styles. The Baumslag Group, strives to inject joy, awe, and appreciation of mathematics in our classrooms; it is our way to honour our dear friend.

*Peggy:*Twenty years later, having left graduate school for a job, the time was right to go back and finish my degree. Gilbert was now on the CUNY GC faculty. I started taking his topics in group theory course, every semester for about 4 semesters. About midway through each lecture, Gilbert would ask us his signature question: Are you feeling weak, or are you feeling strong?" After my course work was complete, I kept sitting in on his courses along with other members of the Baumslag Group. There were often more unregistered students than registered!

*Marcos:*Gilbert was the embodiment of generosity. He loved discussing and sharing ideas with everyone. In the spring of 2007, Gilbert taught a course in finitely generated solvable groups at the CUNY Graduate Center. He gave his notes to some members of the Baumslag Group to organise them, fill in details, produce examples, expand them, and add our own thoughts. This is how I came to learn about semidirect products, wreath products, HNN-extensions, and, of course, finitely generated solvable groups. The notes eventually became a book. This is an example of who Gilbert was. He was all about offering projects, stimulating ideas, and teaching wonderful mathematics. He did all this because he enjoyed it, with boundless energy, and a remarkable sense of humour.

*Anthony:*Following that first class with Gilbert, I started reading some of his papers, even though I could not fully understand most of them. I met with Gilbert several times afterwards, and he directed me to read a particular paper, "On generalised free products." This was the beginning of Gilbert becoming my unofficial advisor. A year or two later, I was still unsure whether he was my official advisor. I did not wish to disrupt the momentum we had established; we had been men at work. One afternoon when Gilbert was speaking with one of his colleagues I overheard him saying, "my student Anthony"; then I knew I was officially Gilbert's student.

*Steve:*Gilbert had a way of explaining difficult subject matter that was understandable for all of the students. He made the course enjoyable by throwing a few jokes around here and there. And he would always ask the audience if they had questions. One day, I met with Gilbert after class to ask for guidance. At that time, I was interested in nilpotent groups and exponential A-groups. Little did I know that Gilbert wrote a book on nilpotent groups. This was certainly meant to be! Gilbert took me under his wing and gave me a thesis project.

**3. Doing Mathematics**

*Anthony:*Gilbert Baumslag was a very kind, generous, and loving teacher and advisor. I met with Gilbert, religiously, every week for about 5 years. During my last year of meeting with Gilbert, I was having some difficulty completing my PhD thesis problem. He consoled and encouraged me by saying "There are lots of people out there with PhDs in Mathematics. I know that you know much more than many of them." That statement, coming from Gilbert, bolstered my self-confidence. Gilbert then left for Europe for 3 months. During his absence, I worked with renewed vigour and inspiration. Just before Gilbert returned, I solved and completed the problem. Gilbert will always have a special place in my heart. I am extremely grateful to have spent some time on this earth with him.

*Marcos:*Interacting with Gilbert was electric. His sheer passion for mathematics and for teaching sustained me throughout graduate school and sustains me still. I often remember those Friday evenings, on our way to dinner after the Group Theory seminar, walking around Manhattan with Gilbert and other seminar goers. Gilbert would discuss mathematics in the street, over dinner, everywhere; always motivated by deep intellectual curiosity. His presence is still felt.

*Peggy:*Gilbert became my advisor for my PhD thesis. It took me forever. Gilbert continues to be my inspiration for mathematics to this very day. Because of Gilbert, Marianna and I joined forces to write a book on "A Sampling of Remarkable Groups." When I lay awake at night struggling with some problem, I always thought, "What would Gilbert do?"

*Marianna:*I had the privilege of working as a research assistant for Gilbert at his lab at CCNY for several years and he was also my thesis advisor for a year. He had a knack for knowing exactly what challenge to pose to me. I recall him writing problems on the board in his office (playing with elements from free products with amalgamation) for me to solve. He would leave me to struggle, chat with someone else, then come back to check on me, always with his white ceramic mug in hand (never a disposable cup in sight!) Gilbert generously doled out ideas, matching them to interest and ability. When I told him I had an idea to use methods from quantum computation to solve decision problems in combinatorial group theory, he encouraged me to find a physicist to advise and guide me, which I did. Gilbert continued to guide me mathematically and pushed me to innovate. Peggy and I still argue: "What would Gilbert do?" What we don't argue about anymore is "He must have made a mistake here!"

*Steve:*Friday was the best day of the week, the day that I would go to the GC for a class taught by Gilbert, go to a seminar and sit with Gilbert, and/or discuss group theory with Gilbert. He was a person that you just wanted to hang out with. I can still hear him say "Every finitely generated torsion-free nilpotent group has a poly-infinite cyclic and central series. Make sure that you can prove this in your sleep." After completing my PhD, I had the idea of writing a book on nilpotent groups, written for both students and professors in the field. Gilbert gave me some ideas. He emphasised that it would take a lot of work and a long time to complete. He advised me that no matter how good the book is, there will always be critics. I pursued the project with Anthony and Marcos. During the years that we worked on it, Gilbert helped us fill in some of the missing pieces from the literature. His support, encouragement, and guidance was truly heartening, especially at times when the work became tiresome and endless. The book, entitled 'The Theory of Nilpotent Groups,' was finally published in 2017. Unfortunately, Gilbert passed on before seeing it. It is dedicated to him. I often think about Gilbert and how he has touched the lives of so many people. God bless him.

*Gretchen:*In 2005, I started working with Gilbert and Chuck Miller on a question about metabelian groups, and then we just kept on working. Eventually, we published a paper about intersections of subgroups and then we moved our attention to decompositions of nilpotent groups. We were trying to figure out under what conditions it could be decided whether a given nilpotent group was decomposable. My last meeting with Gilbert and Chuck together was in August 2014. We were able to finally see our way to an algorithm for deciding decomposability in the case of finitely generated torsion-free nilpotent groups. Gilbert got his diagnosis that month, and was gone within 2 months. It turned out there were quite a few details yet to be worked out, and the work was finally published in 2016. Gilbert was a friend. We didn't talk about personal stuff that much. But Gilbert knew when I was going through difficult personal stuff. He got it, he just got it that life is hard sometimes and that's the way it is, and knowing that Gilbert got it was always a great solace to me. I really don't know how I got so lucky to have been able to work with Gilbert for almost 20 years. It is certainly one of the things in this life for which I feel most grateful. Maybe it wasn't luck, but rather Gilbert's incredible generosity. He gave to so many of us, regardless of our academic stature, that which was most precious: his treasure trove of questions, his time, and his serious attention.

**4. Ilya Kapovich**

The four years (1992-1996) that I was a student of Gilbert Baumslag at the CUNY Graduate Center, formed and defined me as a mathematician. Gilbert played a crucial role in this process. I learned many invaluable lessons from Gilbert during that time: How to think about mathematics, how to write mathematics, how to present and talk about mathematics, how to teach mathematics, and ultimately, how to be a mathematician. Gilbert gave me the time and the freedom necessary to discover what kind of mathematics most agreed with my talents and inclinations, and helped me chose a project with the right mix of geometry, algebra and combinatorics. I also learned from Gilbert that every graduate student is different, and there is no single formula or standard for advising them.

My most intense mathematical interactions with Gilbert occurred when I was working on my first several papers. The process of writing the first of these papers, on small cancellation theory, was particularly agonising (as I thought at the time). I must have gone through a hundred drafts of just the Introduction. After giving each draft to Gilbert, I would get it back densely covered in red ink comments. Every time I thought to myself: Is this pain ever going to end? Gilbert would always smile and say: "Ilya, don't worry, we will get there, it is getting better!" Looking at that paper now, I think that Gilbert should have made me go through a hundred iterations more! He worked hard to break some of my bad writing habits and to replace them by good ones. He taught me the importance of mathematical discipline and clarity, of avoiding complicated notation, overly long (as he said, Tolstovian) sentences and paragraphs, of keeping the main arguments short, clear, and well articulated, of breaking long proofs into shorter lemmas and propositions, and so on. These lessons define my mathematical writing style to this day, and I have tried to impart them on my own students.

Gilbert taught me to direct my writing toward graduate students and mathematicians just entering the field, and not worry too much about trying to impress the experts and the snobs. A crucial conversation, also from the time of writing that small cancellation theory paper, concerned how to decide which mathematics is important. I told Gilbert that I was concerned that in my 20-plus page paper the crucial argument boiled down to a single page with a couple of pictures. Gilbert said: "Ilya, don't worry!" He said that he would tell me a big mathematical secret, which was that in any paper, no matter how long, there is usually a single short place where a crucial argument occurs. If a paper has two such places, then the paper is uncommonly good, and if it has three then, as he put it, "the paper is

*Inventiones*level work."

I learned from Gilbert that at the end you have to trust your own intuition about what kind of mathematics is interesting and important, and not be afraid to pose your own questions and problems. While I was struggling with the Introduction for one of my papers, Gilbert showed me the paper that he was working on the time, called "Musings on Magnus." When I read his Introduction, I was blown away: Not only did he use the pronoun "I" twice in the opening sentence, the first word of that sentence was "I." At the time, I could not imagine doing something so bold and brave in a mathematical paper. I learned from Gilbert the importance of storytelling in writing and presenting mathematics, and in my own papers I have been trying to tell stories ever since then. So far, I have given Gilbert seven "mathematical PhD grandchildren," with two more on the way. In part through them, Gilbert's mathematical legacy and his genius continue to live on. I am proud and honoured to have been Gilbert Baumslag's student.

**5. Delarm Kahrobaei**

I graduated in 2004 under direction of Distinguished Professor Baumslag, writing a thesis related to residual solvability. Afterwards, I went to the University of St Andrews (UK) as an assistant professor. Then for 12 years I was at CUNY, where I got my Full Professorship in 2015, and became a faculty member at the PhD Program in Computer Science. I am currently the chair of cyber security at the University of York (UK) in the Department of Computer Science. I also hold an adjunct professorship position at NYU, Department of Computer Science.

Gilbert Baumslag really helped me to pursue my passion to be a mathematician and computer scientist. He gave me encouragement and support. He also helped me to become a better person.

He encouraged me to dream big, and some of my major accomplishments in part are due to him. He told me examples of successful women mathematicians such as Hanna Neumann, who proved great theorems while she was raising children. He was great at fostering an environment that created opportunities for other people. For example, he organised many conferences and seminars at City College and the Graduate Center focused on many aspects of group theory and some in cryptography. At these conferences/seminars, I met my key collaborators such as Vladimir Shpilrain. Following the model put forward by Gilbert, I have since organised numerous conferences including many which that were designed to encourage women mathematicians and computer scientists.

He also encouraged me to take on leadership roles. For instance, he inspired me to create and be a director of York Interdisciplinary Centre for Cyber Security (University of York, UK, www.cs.york.ac.uk/security).

He was a superb lecturer and mentor. The first conference talk I ever gave was at the Albany Group Theory Conference. He coached me for over a month. He said I am a mathematical descendent of Philip Hall and I must give beautiful lectures. Scientifically I learned a lot from his mentorship to pay attention to details and also his interests in the interdisciplinary fields between combinatorial algebra and cryptography. This made me define a new career path to be a better scholar. He had many students, and learning from his mentorship skills, I have had 8 PhD students (8 academic grandchildren for him), whom all graduated with degrees in mathematics and computer science and have gotten prestigious positions both in academia and industry.

I will miss him very much.

Last Updated February 2023