JOURNAL OF MODERN DYNAMICS WEB SITE: http://www.math.psu.edu/jmd VOLUME 2, NO. 1, 2008, 1–5

MARGULIS

G. DANIEL MOSTOW

This volume is devoted to the mathematical works of Gregory Margulis, and their influence. The editors have asked me to write about aspects of Margulis’ career that might be of general human interest. I shall restrict myself to experi- ences that I personally have witnessed. Thus, this will be one colleague’s obser- vations of Grisha’s life in the USSR and in the USA. I first encountered Margulis’ mathematical power when I read his 1968 joint paper with David Kazhdan proving the existence of nontrivial unipotent ele- ments in non-co-compact lattice subgroups of semisimple Lie groups. This the- orem had been conjectured by A. Selberg and was of special interest to experts in the theory of algebraic groups. In the case of arithmetic lattices, it was equiv- alent to a well-known Godement conjecture which was proved independently in 1960 by Borel–Harish Chandra and Mostow–Tamagawa. In the summer of 1970, I hoped to meet Margulis at a conference in Budapest arranged by Malyusz, a Hungarian student of I. M. Gelfand. This conference was the first opportunity for many western researchers in theory to discuss mathematics with some of their counterparts in the USSR. In addition, there was the good fellowship of the participants, such as flowed with the old wodka brought from Moscow by Ilya Piatetski-Shapiro to his hotel room party for a small group of his Soviet friends and some new friends from the west. (I didn’t keep a diary, and do not remember all who were there, but A. Andrianov of Leningrad and R. Godement of Paris were among the guests.) At the conference, I spoke about the strong rigidity of lattices in Lie groups of R-rank ≥ 2 and for SO(n,1), n > 2 (...n > 2) a result very closely related to Margulis’ 1974 paper submitted to the Vancouver Congress. That is, let Γi be a lattice in a semisimple group Gi (i = 1,2) and let ϕ be an isomorphism of Γ1 to Γ2. Let ϕ∞ denote the induced map of the Furstenberg–Satake boundaries. Then ϕ∞ is Γ1 equivariant and defines an isomorphism of G1 to G2 extending ϕ. The central new idea in my method was to study the map ϕ∞ via its geomet- ric background and to prove that it is smooth. I was pleased to get an excited reaction from Gelfand, who announced to his Soviet colleagues that they would study my talk in his Moscow seminar. Following the 1970 International Congress of at Nice, I served as chair of the IMU Nominating Panel which was charged with selecting speakers for the 1974 Vancouver Congress on Algebraic Groups and Discrete Subgroups.

Received August 12, 2007. 2000 Mathematics Subject Classification: Primary: 01A70. Key words and phrases: Margulis.

1 ©2008 AIMSCIENCES 2 G.DANIEL MOSTOW

The panel members were A. Andrianov of Leningrad, F.Bruhat of Paris, G. Harder of Bonn, M. S. Ragunathan of Bombay, and T. Springer of Utrecht. Margulis was among the leading choices; he had proved the arithmeticity of irreducible lat- tices containing a nontrivial unipotent element in a semisimple Lie group of R- rank ≥ 2. All of the panel members voted for Margulis, except for Andrianov, who voted “no” because he claimed that Margulis’ result was based on an earlier Margulis paper that contained an error. But Andrianov did not identify the specific paper that contained the error. However, most of the panel members knew which pa- per of Margulis contained an error, and also knew that the error had been fixed. I communicated that information to Andrianov and he raised no further objec- tion to the panel about Margulis. In our panel report to IMU Consultative Committee, Margulis was on our list of selected speakers. Unfortunately, that was not the final outcome. S. V. Jablon- ski, the USSR member of the IMU Consultative Committee, acted as an agent of the USSR National Committee for Mathematics, contrary to the IMU code of serving as an independent individual. Jablonski tried to reject all of the se- lected speakers from the USSR not approved by the USSR National Committee for Mathematics. (At that time, disapproval seemed to be inevitable for Jews.) Jablonski found some way to persuade the Consultative Committee to remove Margulis from the list of invited speakers in the section on Algebraic Groups and Discrete Subgroups. Fortunately, he could not find grounds for removing Mar- gulis from the section on Differential Geometry. I remember hoping, in 1974, that Margulis would receive permission from the USSR National Committee on Mathematics to attend the Vancouver Congress and to present his paper. His talk was scheduled for the first day of the Congress, but he was not there. On that first day, at dinner, R. D. James, the chairman of the Organizing Committee on Local Arrangements, handed me a typescript in English of the paper that Margulis had been scheduled to deliver. I read the title: “On Discrete Groups of Motions on Spaces of Nonpositive Curvature,” which was not very informative. As I read further, I realized that this was a landmark paper! This paper settled a question of C. L. Siegel suggested by a construction of G. Giraux: Are all lattices constructable arithmetically? Before 1974, Margulis had such a result for non-cocompact lattices in Q-alge- braic fields of R-rank at least 2. His proof exploited the Kazdan–Margulis The- orem on the existence of unipotent elements 6= 1 in such lattices. But, in a co- compact arithmetic lattice, there is no such element by the Godement criterion. Hence unipotent elements are useless for proving arithmeticity of cocompact lattices. Therefore, a new idea was needed for the case of cocompact lattices. The new idea was to prove the smoothness of the above map ϕ∞ of Fursten- berg–Satake boundaries via measure theory rather than geometry. Take G2 = SL(n,K ), where K is a nondiscrete locally compact field. Margulis’ proof of the measurability of ϕ∞ with respect to Haar measures on G1 and G2 works when ϕ is a homomorphism subject to mild hypotheses that are satisfied in the cases of

JOURNAL OF MODERN DYNAMICS VOLUME 2, NO. 1 (2008), 1–5 MARGULIS 3 interest. Coming into play is the noncommutative Oseledets Ergodic Theorem which generalizes the Birkhoff Ergodic Theorem. The upshot of this measure- theoretic method is the theorem: THEOREM. Let G be a semisimple algebraic R-group and let Γ be an irreducible 0 lattice in GR. LetK bealocalfieldandlet ϕ: Γ → SL(n,K ) be a homomorphism with ϕ(Γ) absolutely irreducible. Assume that the Zariski closure ϕ(Γ) is simple and Zariski-connected. Then 1. If K is neither R nor C, then ϕ(Γ) is relatively compact in the K-topology of SL(n,K). 2. If K = R or K = C, and if ϕ(Γ) is not relatively compact in the K-topology, then ϕ extends to a rational representation of G. I named this result the Margulis Superrigidity Theorem. The arithmeticity of irreducible lattices in semisimple Lie Groups of R-rank ≥ 2 follows in a few lines from this Superrigidity Theorem. In the following letter I describe the reception of this paper of Margulis.

September 4, 1974 Professor G. A. Margulis Institute for Transmission of Information Corpus 2 8 Avia Motornaya Street Moscow 324, USSR

Dear Professor Margulis,

Thank you for your letter of August 22 which I found upon my return from Vancouver. I had wanted to write you in any case.

After being in Vancouver several days, I found that some copies of your English manuscript were being circulated. Unfortunately, I did not receive a copy until after your scheduled talk on August 22. However, realizing the importance of your paper, I presented it at a special seminar on August 28 at 16:35 -- at the time of Kazdan’s scheduled talk, after it was clear that Kazdan had not arrived at the Congress.

You may be pleased to learn that the great enthusiasm which I had for your outstanding work was shared by the large audience, particularly by M. Atiyah of Oxford University, A. Borel of the Institute for Advanced

JOURNAL OF MODERN DYNAMICS VOLUME 2, NO. 1 (2008), 1–5 4 G.DANIEL MOSTOW

Study, and P. Deligne of I.H.E.S. Perhaps some of your Soviet colleagues have already conveyed to you the very warm appreciation for your work displayed at the end of the seminar.

With kindest regards,

G. D. Mostow

The enthusiasm referred to above consisted of gasps of surprise and admiration. In addition to introducing the new idea described above, he had to overcome many technical difficulties in order to prove that the hypotheses of the Oseledets Ergodic Theorem were satisfied. Four years later, at the 1978 International Con- gress of Mathematicians in Helsinki, Margulis was awarded the in an impressive ceremony attended by over 2,000 mathematicians and the fami- lies of three of the four Fields medalists. Unfortunately, Margulis and his family were not able to enjoy this great honor. He was, in fact, in Leningrad, not far away from the meeting, waiting in vain for permission to attend the Congress. In subsequent years, L. Carleson of Sweden, President of the IMU, made sev- eral efforts to arrange a suitable presentation ceremony in the USSR, but he was not successful. Eventually, L. Carleson gave up trying and arranged to award the medal to Margulis in Bonn. The sharp contrast between the respected place of honor that Margulis en- joyed in the mathematical community at large and the lowly place he occupied in the Soviet mathematical establishment presented a painful contrast. I had exchanged letters with Margulis before the Congress to arrange to meet him in Helsinki in case he was given permission to attend and to meet in Leningrad if he was not able to at- tend. After the Congress, my wife and I traveled to Leningrad and met him, as we had planned, in front of the Hermitage. Margulis graciously took my wife and me to many interesting sights in and around Leningrad during the several days that we stayed there, and, in the following week, in Moscow. In order to visit us at our hotel, he had to call me from a public telephone booth near the hotel so that I could come down to the front door and escort him past the guard. Whenever he came to Courtesy of Mathematisches my room, I had to keep in mind that our conver- Forschungsinstitut Oberwolfach sation was being monitored. Professionally, Margulis could not get a professorial position at Moscow Uni- versity, even after winning the medal. In fact, even his efforts to defend his doc- toral dissertation at Moscow University were fruitless. He had a sinecure job at

JOURNAL OF MODERN DYNAMICS VOLUME 2, NO. 1 (2008), 1–5 MARGULIS 5 the Institute for the Transmission of Information, which did have the merit of giving him much free time for mathematics. It was not until 1983 that Margulis was allowed to defend his doctoral disser- tation at the Minsk Institute of Mathematics, whose director, V. Platonov, was in- tent on establishing links with West European and American mathematicians. In later years Platonov was President of the Belarusian Academy of Sciences, served as an adviser to Gorbachov and had a direct phone line to him. In 1990, after Margulis’ only child graduated from high school in Moscow, Margulis emigrated to the U.S. He visited and received job offers from the Uni- versity of Chicago, Harvard, Princeton, Stanford, and Yale. He decided upon Yale, where he has a position to this day. His life became altogether different. Along with the place of honor to which he was elevated came a host of new responsi- bilities: teaching undergraduates and graduate students, directing seminars that prepared its participants to work with him for their PhD dissertations, serving as chair of the mathematics department, and serving on university advisory com- mittees to the President. He never declined to be of service when asked, despite his primary commitment to research. From the fact that he performed his responsibilities with distinction, one can discern that he enjoys these activities, so far removed from his role in Moscow. Nevertheless, he still retains the same modesty that was his nature from his youth. The life that he now enjoys is rich spiritually as well as materially. Grisha’s colleagues have been filled with pleasure to see him win the many honors that he has merited.

G.DANIEL MOSTOW : Department of Mathematics, , PO Box 208283, New Haven, CT 06520-8283

JOURNAL OF MODERN DYNAMICS VOLUME 2, NO. 1 (2008), 1–5