The B. B. Newman Spelling Theorem

This is a guest post by Carl-Fredrik Nyberg Brodda, a recent Masters student at St Andrews and currently a PhD student at the University of East Anglia. The story has personal resonance for me, because it turns out that B. B. Newman was a student at the University of Queensland at about the same time I was, though at a campus 1500km away from Brisbane where I studied – so perhaps I can be forgiven for never having met him. Anyway, here is the extraordinary story.

The full version of this can be found on the arXiv.

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A group presentation is a very compact and natural way to define a group, and can carry a great deal of information about the group it presents. For example, if we take the presentation ⟨a, b | ab=ba⟩, we can directly read that the group it presents is 2-generated, abelian, and torsion-free, and hence must be isomorphic to Z×Z. Beyond compactness, group presentations have a key selling point: they allow us to ask questions about presentations of groups of a combinatorial nature, and, by extension, ask such questions about the groups themselves. A typical example of such a question is asking about the structure of groups defined by presentations ⟨A | w=1⟩ with only a single defining relation, commonly called one-relator groups. Examples of one-relator groups include free groups, the fundamental groups of closed surfaces, and the famous Baumslag-Solitar groups ⟨a, b | b⁻¹a^mb=aⁿ⟩ for m,n non-zero integers. There is a natural partition for one-relator groups into those that have elements of finite order, the torsion case, and those that are torsion-free.

It is known that a one-relator group has torsion if and only if the defining word is a proper power of some other word, i.e. the only time that one has torsion is precisely when one expects it. A common adage in the theory of one-relator groups is that the torsion case is generally more well behaved than the torsion-free. This is in great part because of the following important result from 1968.

Theorem (The B. B. Newman Spelling Theorem)
Let G = ⟨A | Rⁿ=1⟩ be a one-relator group with torsion such that R is cyclically reduced and not a proper power. Let w∈F(A) be a word representing the identity element of G. Then w contains a subword u such that either u or u⁻¹ is a subword of Rⁿ, and such that the length of u is strictly more than (n−1)/n times the length of Rⁿ.

This is similar to a spelling theorem obtained by Dehn for fundamental groups of closed surfaces, and to Greendlinger’s Lemma in small cancellation theory. There are numerous consequences of the theorem for one-relator groups with torsion, with the most important that it shows that such groups are word-hyperbolic, in the sense of Gromov, paving the way for the use of methods from geometric group theory to one-relator groups with torsion. However, seemingly no information about the history of the theorem or of B. B. Newman was available online, which led me to investigate this matter for myself.

The spelling theorem is originally from 1968, and the proof first appears in the PhD thesis of B. B. Newman, at the University of Queensland. However, this thesis was not available anywhere online, and neither was any information as to what two names the Bs were abbreviating. And so, writing to the university library, the hunt was on. Queensland knew some details about Bill Bateup Newman, born in 1936. At Queensland, he had there received an MSc and a BSc, but there was no sign of a doctoral degree, nor any sign of a thesis, nor even any record of who might have supervised Bill. They did have his master’s thesis, however, which was entitled Almost Just Metabelian Groups. There, in the preface, Bill had thanked his supervisor, a Dr M. F. Newman. After some digging, I wrote to now Prof Emeritus Michael Frederick Newman, at the Australian National University in Canberra. He knew some more: Bill’s PhD supervisor had been Gilbert Baumslag, one of the most influential combinatorial group theorists of the 20th century. This came as quite a shock, since Baumslag had been based most of his academic life at the City College of New York, more or less antipodal to Queensland.

Nevertheless, I wrote to New York, and received an email a few days later. However, this email was not from New York. Out of nowhere, I had received an email from B. B. Newman; I learned much later that Mike Newman had contacted Bill, finding his email through a colleague of a widow of a former colleague’s (?!) of Bill’s. Now long since retired, Bill filled in the pieces of the picture I was missing. His doctoral studies had started at Queensland, but not at the main campus in Brisbane. Instead, he had been based in Townsville, almost a thousand miles away, at the University College of Townsville, a college of Queensland. By 1968, he had finished writing his thesis, and was ready to graduate by 1969. Around that time, however, the college officially became James Cook University, the second university in Queensland; hence Bill was given the option to graduate with a degree from either Queensland or James Cook, and chose the latter. This explained why Queensland did not have a copy of his thesis, as no copy was ever submitted there.

As to the matter of his supervision, it was sporadic – Bill recalls a story of how Baumslag sent him a letter written on a German hotel letterhead, told him he was writing from England, and had two weeks later posted the letter from South Africa. It was enough that even Bill himself was not entirely certain on the matter. He thought it might have been Mike Newman initially, which then changed when Baumslag visited Australia in 1964: Baumslag agreed to supervise Bill, but disappeared almost as soon as he had appeared. Not long thereafter, Baumslag sent Bill a copy of a draft of a chapter on one-relator groups from a forthcoming book on combinatorial group theory by Magnus, Karass, and Solitar, and, in Bill’s own words, this was “the most useful help [Baumslag] provided”. This prompted the fruitful investigation into one-relator groups that would culminate in the Spelling Theorem.

After spending some time working with one-relator groups, Bill was due for a sabbatical leave, and Baumslag was in 1967 able to set up a lectureship for him at Fairleigh Dickinson University in Teaneck, New Jersey. This meant that the two mathematicians were able to meet face to face again. At their very first meeting, when explaining how far he had come in proving a Spelling Theorem, Bill realised that his proof was correct. Baumslag, enthusiastically, suggested that the theorem be presented at Magnus’ weekly group theory seminar at the State University in Washington Square, and this came to pass. In the audience that day were both Magnus and Solitar, two of the three who had taught Bill extensively about one-relator groups. What was more, one of these was, of course, Wilhelm Magnus, the man who had proven the Freiheitssatz, the most significant result on one-relator groups to date. Unfazed, Bill presented his results, and concluded his presentation. At that point, Magnus had a remarkable reaction. He jumped to his feet, and exclaimed for the entire room to hear: “I don’t believe this! I don’t believe this!”. In Bill’s own words, he was saying that “he could not believe that an unheard of mathematician, from some unknown university in outback Australia, could have come up with these results”. But the proof was correct.

After my correspondence with Bill, there yet remained a major unresolved issue in the recovery of the thesis. I wrote an abridged email to James Cook containing only the essentials of the story up to that point, for fear of the thesis slipping through my fingers in the time it would take them to read the entire story. Due to time differences, I woke up the next morning pleasantly surprised, having discovered that I had been copied into an overnight flurry of emails sent back and forth between the archivists, heads of research, and librarians at James Cook. Then, at last, the final email of the conversation stated that the thesis had been found, alive and well, and I could not have been happier. Shortly thereafter, I received an email containing the scanned thesis. I was able to read through the original proof, just as I had wanted, and I do not believe that I will ever read a proof with as much enthusiasm again. The thesis was later uploaded to James Cook’s online archives, and also passed on to Queensland, so that they may quickly help anyone in the future digging into the same story.

Carl-Fredrik Nyberg Brodda