Remembering John Stallings Danny Calegari and Benson Farb, coordinating editors Introduction/Authors’ Note This document collects reminiscences of John Stallings by some of us who knew him. The contribu- tions have been minimally edited and are supplemented by Stallings’s own “Items from my unknown autobiography”. Photo credits are given where known. Danny Calegari Bestvina-Brady’s discrete Morse theory), and clari- John Robert Stallings Jr. was born fying the work of Serre on trees, especially as it in Morrilton (which John pro- applied to Haken 3-manifolds. In 1983 he wrote nounced “Marrrlton”, rolling the a brief paper, “Topology of finite graphs”, which, r’s), Arkansas, on the 22nd of July besides giving astonishingly short and illuminating 1935, and died at his home in proofs of notoriously opaque algebraic theorems, Berkeley, California, on the 24th anticipated to some extent the theory of automatic of November 2008. He achieved groups and structures developed by Cannon, Thur- early fame as a postdoc with his ston, Epstein et al. Later work was more sporadic alternative proof of the Poincaré but included such important notions as triangles Conjecture in high dimensions of groups and angles between subgroups and con- (obtained only days after Smale’s tinued to be guided by the interaction between (es- breakthrough) using an original pecially 3-dimensional) topology and group theory. method known as engulfing. John John is remembered as much for his anti- Photo credit: Enrico Le Donne, 2006. then switched his attention to John Robert Stallings Jr. establishment attitudes and distinctive personal- 3-manifold topology and geomet- ity as for his mathematics (and sometimes for the ric group theory, and transformed interaction between the two, such as in his famous both. He brought combinatorial methods from paper “How not to prove the Poincaré Conjecture”); group theory to 3-manifold theory, which at the some of this is captured in the reminiscences time was preoccupied with the point-set methods below. He was always a champion of the original- of Bing and Wilder — giving new and cleaner proofs ity and brilliance of teenagers and young people, of Papakyriakopoulos’s fundamental discoveries, keeping in touch with many young correspondents characterizing fibered 3-manifolds by using the even in his seventies and taking great delight in fundamental group, and introducing important the music and antics of modern enfants terrible new methods (e.g., plumbing) in knot theory. Con- such as Eminem. Both aspects of his personality versely, he brought geometry and topology back are revealed in the following excerpt from his to group theory, reinvigorating a field that had introduction to a talk at Barry Mazur’s birthday forgotten the perspective of Dehn and Nielsen, conference in 1998: and laid the groundwork for the later radical in- novations of Thurston and Gromov: characterizing There are some lessons to be learned groups with infinitely many ends (an achievement from this story. One that appeals to me for which he was recognized with the Cole Prize is that Barry broke the rules; he didn’t in 1965), constructing an example of a finitely have a Bachelor’s degree, he didn’t stay presented group with infinite-dimensional H3 quietly in Princeton for three years. (an example which became the cornerstone of Even Douady broke the unspoken rule that one is supposed to wear clothes Danny Calegari is Richard Merkin Professor of Math- under the gown. All those rules, BA ematics at Caltech. His email address is dannyc@its. diploma, residence in Princeton, wear- caltech.edu. ing shoes, being nice to chairmen and 1410 NOTICES OF THE AMS VOLUME 56, NUMBER 11 deans, being a team player—basically I a Ph.D. from Princeton. His thesis gave a new hate those rules, even though I give in topological proof of an old important theorem of and adjust to them. It makes me feel Grushko in combinatorial group theory. This was good to see a rule-breaker succeed! the first indication of how Stallings would trans- form combinatorial group theory by introducing And, teenagers are creative and amaz- powerful topological methods. ing. Barry’s theorem could be compared Stallings definitely followed his own path in to Opus One of some great composer. mathematics. He was known for his great origi- For instance, Gustav Mahler, at the nality; he most often came up with completely age of 17, wrote a long poem, and original ideas rather than following up on the ideas set it to music by the age of 19. This of others. His work often inspired a great flurry cantata, “Das klagende Lied”, is rarely of activity by other mathematicians, who would performed in its original form, because develop Stallings’s methods. There are important Mahler “improved” it later on. I recently cases, though, including his Ph.D. thesis, where heard the original work in Berkeley, Stallings reunderstood someone else’s theorem in played by the Berkeley Symphony Or- a completely new way, which would open the door chestra under Kent Nagano, with a to new developments on an old topic. large chorus, six vocal soloists, two Stallings made fundamental contributions to a of which were from the San Francisco number of different areas of mathematics, from Girls Choir, and an offstage band from combinatorial and geometric group theory to knot Hayward State University that played theory to the theory of 3-dimensional manifolds. in a different key and time-signature. Indeed, he was one of the main forces responsible It had its adolescent moments, but it is for tying these areas together. truly a great work of art. Just as Barry’s Stallings’s ideas were usually remarkably simple sphere theorem was. but remarkably deep. They seemed to grow out of his incessant doodling. He doodled all the time, Those of us who are antiestablishmentarians especially when he attended talks. One example at heart or who met him later in his life probably of a simple but powerful idea is Stallings’s beauti- remember John more for his laconic wit, his pre- ful paper “Topology of finite graphs”. How many dilection for Thai food or for beer and pizza after people could write a deep paper with that title? In seminars (at La Val’s or Jupiter), his fondness for this paper Stallings took our understanding of free reading “Zits” in the morning paper (usually in groups to a new level, along the way making older, Brewed Awakenings), his impossible turn of speed difficult theorems look almost obvious. around about the fourth mile of a hike to Point Stallings was greatly respected as a mathemati- Reyes, his long gossipy monologues while driving cian. One acknowledgment came in the form of his out to a seminar at Stanford, his provocations and being awarded the Cole Prize for his now famous his unconventional behavior (tearing up a Bulletin theorem on ends of groups: a group splits as an article on “Making proofs without modus ponens” amalgamated product over a finite group if and with mock rage; speculating on the likelihood that only if it has at least two ends. Here is a paradig- Alex Gottlieb would fall off the balcony outside matic example of how fine algebraic structure can 1015 Evans) than for his mathematical insights, be determined by coarse, large-scale information. which were rare in his later years but seemed rarer No wonder this theorem is a cornerstone of geo- still because of his modesty and high standards. The truth is, we have internalized his mathematics metric group theory. so that it seems less like a reflection of John and Everyone loved Stallings. He was always gener- more like a fact of nature. He felt more like a fellow ous with his time and with his ideas. He treated graduate student than one of the exalted faculty. students with the same respect he did colleagues, For generations of graduate students at Princeton, indeed with more respect. Stallings had a dislike of Berkeley, and elsewhere, Stallings was larger than authority and made a point of playing by his own life. It is hard to believe that he is gone. rules. He often made fun of authority, but not in a mean way. He made fun of himself all the time. Benson Farb Actually, I can’t think of anything or any person John Stallings was a one-of-a-kind mathematician. (including himself) that Stallings took seriously. Perhaps this is due to the fact that he came out of Stallings was widely known not just for his ge- a small town in Arkansas. He loved to play up his nius but for his sense of humor. As one example, roots, often portraying himself as a kind of naive he wrote an entire paper in the universal language country bumpkin. Of course, he ended up getting Interlingua. One of his famous papers on one of the most famous mathematical problems, “How not Benson Farb is professor of mathematics at the University to prove the Poincaré Conjecture”, begins: “I have of Chicago. His email address is [email protected]. committed—the sin of falsely proving Poincaré’s edu. Conjecture. But that was in another country; and DECEMBER 2009 NOTICES OF THE AMS 1411 besides, until now beautiful paper on the loop theorem. By the time I no one has known was done I had a sense of what 3-manifold theory about it.”1 was about and how closely connected to group Stallings was in- theory it was, and I felt that this was my area of credibly generous research, even though I had not yet made contribu- with his time. He tions of my own.