# Jacques Tits

### Quick Info

Born
12 August 1930
Uccle, Belgium

Summary
Jacques Tits is a Belgium-born French mathematician who worked on group theory and geometry.

### Biography

Jacques Tits was born in Uccle, on the southern outskirts of Brussels. His parents were Léon Tits, who was a professor, and Lousia André. Jacques attended the Athénée of Uccle and then studied at the Free University of Brussels. His thesis advisor in Brussels was Paul Libois, and Tits graduated with his doctorate in 1950 having submitted his dissertation Généralisation des groupes projectifs basés sur la notion de transitivité . From 1948 to 1956 he was funded by the Belgium Fonds National de la Recherche Scientifique.

Tits' first papers, following the work he had undertaken for his doctoral dissertation, were on generalisations of triply transitive groups. He published a two part paper Généralisations des groupes projectifs in 1949 on this topic generalising the group of one-dimensional projective transformations. Among the results proved were characterisations of projective groups among triply transitive groups. In Groupes triplement transitifs et généralisations (1950), Tits went on to look at generalisations of $n$-tuply transitive groups, defining an almost $n$-tuply transitive group. This generalises the group of collineations of the plane which is almost quadruply transitive. In Sur les groupes triplement transitifs continus; généralisation d'un théorème de Kerékjártó (1951) Tits looked at triply transitive groups of transformations of a topological space using his earlier results which characterised the projective groups among triply transitive groups.

Tits married Marie-Jeanne Dieuaide, a historian, on 8 September 1956. From 1956 to 1962 he was an assistant at the University of Brussels. He was promoted to professor in 1962 and remained in this role at Brussels for two years before accepting a professorship at the University of Bonn in 1964. Among Tits' doctoral students in Brussels we mention Francis Buekenhout who was awarded his doctorate in 1965. In 1973 Tits accepted the Chair of Group Theory at the Collège de France. Shortly after taking up this post, he became a naturalised French subject in 1974. Tits held this chair until he retired in 2000.

The large and important mathematical developments introduced by Tits are far too numerous to cover here in any detail. Perhaps the most important part of his work was the introduction of buildings and this is put into context by Ronan in [3]. We give his summary:-
This paper is an essay on how the development of group theory led to the discovery of various families of simple groups, and how these in turn led to the theory of buildings. In outline the story is this. Galois first used the term 'group' in the technical sense, and found the first simple groups. Jordan, in his famous Traité des substitutions et des équations algébriques , published in 1870, promoted Galois' work and put the theory of groups on a firm foundation. At this time groups were treated as groups of permutations, but other aspects of group theory were soon on the way. Lie visited Paris in 1870 as a graduate student, and went on to create the theory of continuous transformation groups. Killing came to such groups independently, and in 1888 found the classification of the simple Lie groups, using semisimple complex Lie algebras (families A through G). Cartan refined this classification in 1894, correcting some errors in the proofs, and it is now known as the Killing-Cartan classification. The classical families (A through D) soon led to groups over fields other than the real or complex numbers, and a comprehensive study was published by Dickson in 1901. Later he dealt with $E_{6}$ and G, but progress on the others did not occur until after the Second World War. Tits was working on the problem, as was Chevalley, who was a more established mathematician at that time. Chevalley succeeded in 1955, and his paper was soon followed by variations due to Steinberg, Tits, Suzuki, and Ree. During this time Tits was gradually developing the theory of buildings, and his book "Buildings of spherical type and finite BN-pairs" in 1974 produced a fully-fledged theory that has since found many uses. ... we mention some of Tits' early work on buildings, and we discuss the contents of his above-mentioned book concerning buildings of spherical type. Finally ... a later approach to buildings, also due to Tits, is mentioned, and we return at the end to the construction of the exceptional groups of Lie type using building theory.
Through a large number of other important roles, Tits played a major part in mathematical life. For example he was editor-in-chief for mathematical publications at I.H.E.S. from 1980 to 1999. He served on the committee awarding the Fields medals in 1978 and again in 1994. He also served on the committee awarding the Balzan Prize in 1985.

After retiring in 2000, Tits became the first holder of the Vallée-Poussin Chair from the University of Louvain. He gave his inaugural lecture Immeubles : une approche géométrique des groupes algébriques simples et des groupes de Kac-Moody on 18 October 2001. He followed this with three series of lectures on the following topics
(1) Généralités sur les nombres $p$-adiques. Groupes algébriques simples sur les corps $p$-adiques ;
(2) Schémas en groupes à fibre générique simple sur les anneaux d'entiers ;
(3) Réseaux invariants dans les espaces de représentations. Applications algébriques .
In 2008 the Norwegian Academy of Science and Letters awarded the Abel Prize to John Griggs Thompson and Jacques Tits:-
... for their profound achievements in algebra and in particular for shaping modern group theory.
The Press Release gives the following summary of Tits's contributions:-
Tits created a new and highly influential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the algebraic structure of linear groups. The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classification of algebraic and Lie groups as well as finite simple groups, to Kac-Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces. Tits's geometric approach was essential in the study and realisation of the sporadic groups, including the Monster. He also established the celebrated "Tits alternative": every finitely generated linear group is either virtually solvable or contains a copy of the free group on two generators. This result has inspired numerous variations and applications. The achievements of John Thompson and of Jacques Tits are of extraordinary depth and influence. They complement each other and together form the backbone of modern group theory.

### References (show)

1. H Abels, Cantor-Medaille für Jacques Tits, Jahresber. Deutsch. Math.-Verein. 103 (1) (2001), 7-18.
2. F Buekenhout, A Belgian mathematician : Jacques Tits, Bull. Soc. Math. Belg. Sér. A 42 (3) (1990), 463-465.
3. M Ronan, From Galois and Lie to Tits buildings, in The Coxeter legacy (Amer. Math. Soc., Providence, RI, 2006), 45-62.
4. A Valette, Quelques coups de projecteurs sur les travaux de Jacques Tits, Gaz. Math. No. 61 (1994), 61-79.

### Additional Resources (show)

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### Honours (show)

Honours awarded to Jacques Tits

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Written by J J O'Connor and E F Robertson
Last Update September 2009