Jacques Tits

How to use this course

Jacques Tits changed what a group is for. Before him, the simple groups of Lie type were studied through matrices, root systems and long computations, case by case. Tits attached to each such group a geometric object, its building, and showed that the geometry determines the group. The same idea, applied over \(p\)-adic fields, gave Bruhat–Tits theory. His other results include the Tits alternative for linear groups, a simplicity theorem used to prove whole families of groups simple at once, and a construction of the exceptional Lie algebras.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The running example throughout is \(\mathrm{GL}_n\) over a field, where every abstract notion has a concrete meaning in terms of flags of subspaces.

Timeline

Uccle and the university at fourteen, 1930–1948

Jacques Tits was born on 12 August 1930 in Uccle, a commune on the southern outskirts of Brussels. His father, Léon Tits, was a professor of mathematics, and his mother was Lousia André. The boy’s ability showed very early. According to the Abel Prize biography, he could carry out the four arithmetic operations by the age of three, and he skipped several years of school.

He attended the Athénée of Uccle. When he was 13 his father died, and the family had little money. Jacques helped by tutoring pupils about four years older than himself. At 14 he passed the entrance examination of the Free University of Brussels (Université libre de Bruxelles), where he studied mathematics.

His training at Brussels was in geometry, and his teacher Paul Libois was a geometer. The synthetic outlook, which reasons with incidences of points, lines and planes rather than with coordinates, stayed with Tits throughout his career. Even his most algebraic theorems are, at heart, statements about how such objects fit together.

Transitive groups, 1948–1956

From 1948 to 1956 Tits was supported as a research fellow by the Belgian Fonds National de la Recherche Scientifique. His doctoral advisor was the geometer Paul Libois. In 1950, aged 20, he received his doctorate for the thesis Généralisation des groupes projectifs basés sur la notion de transitivité.

The thesis starts from a classical fact. The group \(\mathrm{PGL}_2(K)\) of fractional linear maps \(x\mapsto\frac{ax+b}{cx+d}\) acts on the projective line \(K\cup\{\infty\}\) sharply triply transitively: any ordered triple of distinct points can be moved to any other by exactly one element. Tits asked which groups share this property, and what geometry they carry. His papers of 1949 and 1950 developed the notion of almost \(n\)-tuply transitive groups. A 1951 paper extended a theorem of Béla Kerékjártó on continuous triply transitive groups, bringing topological groups into the subject.

The direction of these early papers is the direction of his life’s work. Tits wanted to recover a group from the geometry it acts on, and a geometry from its group of symmetries. Projective geometry over a field is determined by its incidence structure (the fundamental theorem of projective geometry), and the projective group is determined by the geometry. The question that drove him in the 1950s was how to extend this dictionary from \(\mathrm{PGL}_n\) to all simple Lie groups, including the exceptional groups \(G_2\), \(F_4\), \(E_6\), \(E_7\), \(E_8\).

He received the Prix scientifique interfacultaire L. Empain in 1955. In 1955 Claude Chevalley constructed analogues of all simple Lie algebras over arbitrary fields, including finite fields, in a celebrated paper. Tits had been pursuing the same goal through geometry, and his geometric viewpoint would later explain Chevalley’s groups in a uniform way.

Brussels to Bonn, 1956–1973

In 1956 Tits became an assistant at the University of Brussels, and on 8 September of that year he married the historian Marie-Jeanne Dieuaide. That year, at a colloquium on higher algebra in Brussels, he presented the paper published in 1957 as Sur les analogues algébriques des groupes semi-simples complexes. It contains the first form of what became buildings, and the remark that the symmetric group behaves like a linear group over a field “of characteristic one”, an idea revived fifty years later (Lesson 10).

Tits was one of the first visitors to the Institut des Hautes Études Scientifiques after its foundation in 1958, and the second volume of its Publications (1959) carried his long paper on triality, which introduced generalized polygons. He received the Wettrems Prize of the Belgian Academy in 1958. In 1962 he became professor at Brussels, was an invited speaker at the International Congress of Mathematicians in Stockholm, and published the short note in which BN pairs appear. He also received the François Deruyts Prize that year.

In 1964 he moved to the University of Bonn, where he stayed for a decade. His 1964 paper Algebraic and abstract simple groups in the Annals of Mathematics proved simplicity for the rational points of isotropic simple algebraic groups in great generality, and the same year he showed that the derived subgroup of the Ree group \({}^2F_4(2)\) is a new simple group, now called the Tits group. With Armand Borel he wrote Groupes réductifs (1965), the standard account of the structure of reductive groups over arbitrary fields. In Bonn he supervised doctoral students including Jens Carsten Jantzen and Karl-Otto Stöhr. His first doctoral student, Francis Buekenhout, had finished at Brussels in 1965.

Tits became an honorary member of the Bourbaki group. He is also responsible for familiar names: he popularized the work of H. S. M. Coxeter and introduced the terms Coxeter group, Coxeter graph and Coxeter number.

The Bonn decade ended with two landmark papers in 1972: the Tits alternative (Lesson 08) and, with François Bruhat, the first part of the theory of reductive groups over local fields (Lesson 07).

Coxeter groups and BN pairs

For the general reader. The symmetries of a regular polygon or polyhedron are generated by reflections, and a few rules about how pairs of reflections combine determine everything. Groups defined by such rules are Coxeter groups. Tits saw that every simple group of Lie type, from \(\mathrm{SL}_n\) over any field to the exceptional groups, contains a hidden Coxeter group, and he isolated the few axioms that make this work: the BN pair.

Coxeter systems

A Coxeter matrix on a finite set \(S\) is a symmetric matrix \((m_{st})\) with \(m_{ss}=1\) and \(m_{st}\in\{2,3,\dots,\infty\}\) for \(s\ne t\). The Coxeter group is

and \((W,S)\) is a Coxeter system. The length \(\ell(w)\) is the shortest length of a word in \(S\) representing \(w\).

BN pairs

Here \(BwB\) means \(BnB\) for any \(n\in N\) mapping to \(w\); it does not depend on the choice because \(T\subseteq B\).

Proof sketch. Axiom 3 shows that \(\bigcup_w BwB\) is stable under left multiplication by \(B\) and by each \(s\), hence by \(G=\langle B,N\rangle\); so it is all of \(G\). For disjointness one proves \(BwB=Bw'B\Rightarrow w=w'\) by induction on \(\ell(w)\), using axiom 4 to show that \(sBsB\) meets \(BsB\), and hence that \(BsBwB=BswB\) whenever \(\ell(sw)>\ell(w)\).

The example \(\mathrm{GL}_n(K)\)

Take \(B\) the upper triangular matrices and \(N\) the monomial matrices (one non-zero entry in each row and column). Then \(T\) is the diagonal subgroup, \(W=N/T\cong S_n\), and \(S\) consists of the adjacent transpositions \(s_i=(i\ i{+}1)\), with \(m_{s_is_{i+1}}=3\) and \(m_{s_is_j}=2\) for \(|i-j|\ge2\). The Bruhat decomposition is the statement that Gaussian elimination by row operations within \(B\) reduces any invertible matrix to a permutation matrix. The parabolic subgroups are the block upper triangular groups, one for each composition of \(n\).

The simplicity theorem

For the general reader. Simple groups are the atoms of group theory: they cannot be broken into smaller pieces. Proving that a family of groups is simple used to require a separate argument for each family. Tits gave one argument that works for all groups with a BN pair at once, provided a few easily checked conditions hold.

Proof

Let \(H\trianglelefteq G\). Then \(HB\) is a subgroup containing \(B\), so \(HB=P_J\) for some \(J\subseteq S\) by Lesson 04. Since \(H\) is normal, one checks that \(J\) commutes elementwise with \(S\setminus J\): for \(s\in J\), \(t\notin J\), conjugating a representative of \(s\) in \(H\) by \(t\) stays in \(HB=P_J\), which forces \(tst\in W_J\), and this happens only if \(s\) and \(t\) commute. Irreducibility leaves two cases.

If \(J=\varnothing\), then \(H\subseteq B\); being normal, \(H\subseteq gBg^{-1}\) for every \(g\), so \(H\subseteq Z\).

If \(J=S\), then \(G=HB=HUT\). Every conjugate of \(U\) lies in \(HU\), because \(gUg^{-1}=hbUb^{-1}h^{-1}\subseteq hUh^{-1}\subseteq HU\) for \(g=hb\) (note \(U\trianglelefteq B\) and \(H\trianglelefteq G\)). So \(G=HU\) by condition 2, and \(G/H\cong U/(U\cap H)\) is solvable. A perfect group with a solvable quotient has trivial quotient, so \(H=G\). \(\square\)

Application

For \(G=\mathrm{SL}_n(K)\), let \(U\) be the upper unitriangular matrices. The conjugates of \(U\) contain all elementary matrices \(I+aE_{ij}\), which generate \(\mathrm{SL}_n(K)\), and \(\mathrm{SL}_n(K)\) is perfect except for \((n,|K|)=(2,2),(2,3)\). The Weyl group \(S_n\) is irreducible, and \(Z\) is the group of scalar matrices of determinant 1. The theorem gives the classical result: \(\mathrm{PSL}_n(K)\) is simple except for \(\mathrm{PSL}_2(\mathbb F_2)\cong S_3\) and \(\mathrm{PSL}_2(\mathbb F_3)\cong A_4\). The same argument covers the Chevalley groups and their twisted forms.

Spherical buildings

For the general reader. A building is a geometric object assembled from many copies of a single tiled sphere, the apartments, glued together so that any two rooms (chambers) lie in a common apartment. For the group of invertible matrices, the rooms are the complete flags of subspaces. Tits proved that buildings of rank at least 3 are so rigid that each one comes from a group, and he classified them all.

Definition

A Coxeter complex \(\Sigma(W,S)\) is the simplicial complex whose simplices are the cosets \(wW_J\), \(J\subsetneq S\), ordered by reverse inclusion. Its maximal simplices (chambers) correspond to the elements of \(W\). For finite \(W\) it triangulates a sphere of dimension \(|S|-1\).

The building of a vector space

For \(\mathrm{GL}_n(K)\), the building is the flag complex: vertices are proper non-zero subspaces of \(K^n\), simplices are chains \(V_1\subset V_2\subset\cdots\), and chambers are complete flags. An apartment is determined by a frame, a decomposition \(K^n=L_1\oplus\cdots\oplus L_n\) into lines; its vertices are the sums of subsets of the \(L_i\). Each apartment is the Coxeter complex of \(S_n\), the barycentric subdivision of the boundary of an \((n-1)\)-simplex. For \(n=3\) the building is a bipartite graph (points and lines of the projective plane), and each apartment is a hexagon: three points in general position and the three lines joining them.

Rank 2: generalized polygons

A building of rank 2 is a bipartite graph. Its apartments are \(2m\)-cycles, where \(W\) is dihedral of order \(2m\), and the axioms say: the graph has diameter \(m\) and girth \(2m\). Tits introduced these generalized \(m\)-gons in 1959. Generalized 3-gons are projective planes. The group \(G_2(q)\) acts on a generalized hexagon, the split Cayley hexagon, and the Ree groups \({}^2F_4(q)\) act on the Ree–Tits octagons.

Rank 2 is different: there are many exotic generalized polygons, including free constructions. Tits and Richard Weiss classified those with the Moufang property in their 2002 book Moufang Polygons; Earlier, Tits (1976) and Weiss (1979) had shown that thick Moufang \(m\)-gons exist only for \(m\in\{3,4,6,8\}\), without assuming finiteness.

Bruhat–Tits theory and trees

For the general reader. Over the \(p\)-adic numbers, the group \(\mathrm{SL}_2\) acts on an infinite tree in which every vertex has \(p+1\) neighbours. François Bruhat and Tits showed that every reductive group over such a field acts on a similar space, an affine building, built from flat Euclidean apartments. Geometry of this space answers algebraic questions, such as the classification of maximal compact subgroups.

The tree

A lattice in \(\mathbb Q_p^2\) is a \(\mathbb Z_p\)-submodule of rank 2, and two lattices are equivalent if one is a scalar multiple of the other. Let \(X\) be the graph whose vertices are classes \([L]\), with \([L]\) adjacent to \([L']\) when representatives satisfy \(pL\subsetneq L'\subsetneq L\).

The neighbours of \([L]\) correspond to the lines in \(L/pL\cong\mathbb F_p^2\), which form \(\mathbb P^1(\mathbb F_p)\), with \(p+1\) elements. The link of a vertex is thus the spherical building of \(\mathrm{SL}_2(\mathbb F_p)\), a general feature of affine buildings. The stabilizer of the standard edge is the Iwahori subgroup, the matrices in \(\mathrm{SL}_2(\mathbb Z_p)\) that are upper triangular mod \(p\), and together with the monomial matrices it forms a BN pair with infinite (affine) Weyl group, as Nagayoshi Iwahori and Hideya Matsumoto had found in 1965.

Proof. For a bounded set \(Y\), let \(r(x)=\sup_{y\in Y}d(x,y)\). In a CAT(0) space \(r\) has a unique minimizer, the circumcentre: if \(x_1,x_2\) both nearly minimize \(r\), the semi-parallelogram law applied at their midpoint \(m\) gives \(d(x_1,x_2)^2\le 2r(x_1)^2+2r(x_2)^2-4r(m)^2\), which forces \(x_1,x_2\) close; completeness gives existence. Apply this to an \(H\)-orbit \(Y\). Since \(H\) permutes \(Y\), it fixes the circumcentre. \(\square\)

A compact subgroup of \(\mathrm{SL}_2(\mathbb Q_p)\) has bounded orbits on \(X\), so it fixes a point, which is a vertex or the midpoint of an edge; as \(\mathrm{SL}_2\) does not invert edges, it fixes a vertex. Hence:

Bruhat and Tits built affine buildings for all reductive groups over fields with a discrete valuation, in two long papers in the Publications mathématiques de l’IHÉS (1972 and 1984). The theory is now basic to the representation theory of \(p\)-adic groups and the Langlands programme. Jean-Pierre Serre’s book Trees develops the rank 1 case; one of its applications is Ihara’s theorem that every torsion-free discrete subgroup of \(\mathrm{SL}_2(\mathbb Q_p)\) is free, proved by letting the group act freely on the tree.

The Tits alternative

For the general reader. A group of matrices generated by finitely many elements is either tame, close to a group whose commutators eventually vanish, or wild, containing a copy of the free group in which no non-trivial word in two generators equals the identity. There is no middle ground. This dichotomy, proved by Tits in 1972, became a model for theorems across geometric group theory.

Ping-pong

Proof. A non-trivial reduced word is conjugate to one of the form \(w=a^{k_1}b^{l_1}\cdots b^{l_r}a^{k_{r+1}}\) with all exponents non-zero, beginning and ending with a power of \(a\). For \(x\in X_b\), apply the letters from right to left: \(x\) moves into \(X_a\), then \(X_b\), alternately, and ends in \(X_a\). So \(w(x)\in X_a\), which is disjoint from \(X_b\ni x\); hence \(w\ne1\). \(\square\)

Outline of Tits’s proof

Pass to the Zariski closure of \(\Gamma\) and its identity component. If that is solvable, so is a finite-index subgroup of \(\Gamma\). Otherwise one reduces to a group whose image in a semisimple group is Zariski dense. Tits then finds, using that \(\Gamma\) is finitely generated, an embedding of the field generated by the matrix entries into a local field \(k\) (such as \(\mathbb R\), \(\mathbb Q_p\) or \(\mathbb F_q((t))\)) in which some element of \(\Gamma\) has an eigenvalue of absolute value different from 1. Taking a suitable representation, he obtains proximal elements: elements acting on projective space with an attracting fixed point and a repelling hyperplane. Two such elements in general position, raised to high powers, play ping-pong near their attracting points.

Consequences

• Growth. A finitely generated linear group has either polynomial or exponential growth. Free groups have exponential growth; virtually solvable groups have polynomial or exponential growth by theorems of John Milnor and Joseph Wolf.
• Amenability. A linear group is amenable if and only if it contains no non-abelian free subgroup. The von Neumann conjecture, false for general groups, holds for linear ones.
• Gromov’s theorem. Mikhail Gromov’s 1981 theorem that groups of polynomial growth are virtually nilpotent uses the Tits alternative to handle the linear groups that appear in its proof.
• Analogues. The alternative was later proved for mapping class groups (Nikolai Ivanov; John McCarthy), for \(\mathrm{Out}(F_n)\) (Bestvina, Feighn and Handel), and for hyperbolic groups.

Exceptional groups and the magic square

For the general reader. The five exceptional Lie groups do not belong to the infinite families of rotations and matrices. Tits found uniform constructions for them using the octonions and Jordan algebras, and he discovered a finite simple group that stands between the regular families and the sporadic groups.

The Freudenthal–Tits magic square

Let \(A\) be one of the normed division algebras \(\mathbb R,\mathbb C,\mathbb H,\mathbb O\), and \(J=H_3(A')\) the Jordan algebra of \(3\times3\) Hermitian matrices over another one, \(A'\). Write \(A_0\), \(J_0\) for the trace-zero parts and \(\mathrm{Der}\) for the Lie algebra of derivations. In 1966 Tits defined a Lie bracket on

Hans Freudenthal had found the square through the geometry of the octonionic planes in a series of papers from 1954 to 1963; Tits’s formula gives it in one stroke. The symmetry of the table is not visible in Tits’s formula, which treats \(A\) and \(J\) differently; later constructions by Ernest Vinberg and by Barton and Sudbery make it manifest.

The Tits group

Rimhak Ree constructed in 1961 a family of groups \({}^2F_4(q)\), \(q=2^{2m+1}\), of order \(q^{12}(q^6+1)(q^4-1)(q^3+1)(q-1)\). For \(q\ge8\) these are simple. For \(q=2\) Tits showed in 1964 that the derived subgroup \({}^2F_4(2)'\) has index 2 and is a new simple group. Because it is not itself a group of Lie type, it is sometimes counted as a 27th sporadic group.

Geometry and the sporadic groups

The 26 sporadic simple groups have no BN pair, but many act on geometries that look locally like buildings: incidence structures whose residues (the structures seen from one element) are generalized polygons or projective spaces. Tits’s 1981 paper A local approach to buildings showed that a chamber system whose rank 3 residues are spherical buildings is covered by a building, and the same local viewpoint was used to construct and study sporadic groups, including the Monster. Tits liked a smaller example: the Janko group \(J_2\), of order \(604\,800\), acts on a graph with 100 vertices, and he noted that its order is the number of seconds in a week.

Kac–Moody groups and the field with one element

For the general reader. Buildings need not be finite or flat. Infinite-dimensional Lie algebras discovered in the late 1960s have groups attached to them, and Tits gave the definitive construction of these groups over every field, together with a new kind of building. At the other extreme, he noticed in 1956 that finite sets behave like vector spaces over a field with one element, an idea that later fed into attempts on the Riemann hypothesis.

Kac–Moody groups

A generalized Cartan matrix \(A=(a_{ij})\) has \(a_{ii}=2\), \(a_{ij}\le0\) for \(i\ne j\), and \(a_{ij}=0\iff a_{ji}=0\). In 1968 Victor Kac and Robert Moody independently attached Lie algebras to such matrices; finite-type matrices give the simple Lie algebras, and affine-type matrices give loop algebras. The Weyl group is the Coxeter group generated by \(s_i\) with relations determined by the products \(a_{ij}a_{ji}\).

In 1987 Tits defined, for every generalized Cartan matrix and every field \(K\), a group \(G_A(K)\) by generators and relations modeled on Robert Steinberg’s presentation of Chevalley groups, and proved that it is uniquely characterized by natural axioms. Such a group carries two BN pairs, \((B_+,N)\) and \((B_-,N)\), with opposite Borel subgroups and the same Weyl group, linked by a refined Bruhat decomposition \(G=\bigsqcup_w B_+wB_-\). The two buildings, together with a codistance between them, form a twin building, a notion Tits developed with Mark Ronan. Over a finite field, suitable Kac–Moody groups give lattices in the product of the two buildings’ automorphism groups, a rich source of examples in geometric group theory.

Tits’s 1957 paper made this observation. From the 1990s Yuri Manin, Christophe Soulé, Alain Connes, Caterina Consani and others tried to build algebraic geometry over \(\mathbb F_1\), hoping to treat \(\mathrm{Spec}\,\mathbb Z\) as a curve over it and transfer André Weil’s proof of the Riemann hypothesis for curves over finite fields.

The Collège de France, 1973–2000

In 1973 Tits was elected to the Chair of Group Theory at the Collège de France in Paris. The Collège then required its professors to hold French nationality, and Belgian law did not allow dual citizenship, so in 1974 he became a French citizen and gave up his Belgian nationality. He held the chair until 2000. Courses at the Collège are open to all and must present new research each year; Tits used them to develop his ideas on buildings, Kac–Moody groups and Moufang polygons, and his yearly summaries in the Collège’s Annuaire are a record of that work.

1974 also brought Buildings of Spherical Type and Finite BN-Pairs (Springer Lecture Notes in Mathematics 386), the classification of Lesson 06, and an invited address at the International Congress in Vancouver. He received the Grand Prix des sciences mathématiques et physiques of the Académie des sciences in 1976 and was elected to the Académie in 1979. He joined the Leopoldina in 1977, the Royal Netherlands Academy and the newly founded Academia Europaea in 1988, the Royal Academy of Belgium in 1991, and the American Academy of Arts and Sciences and the US National Academy of Sciences in 1992.

From 1980 to 1999 he was editor-in-chief of the Publications mathématiques de l’IHÉS, among the most selective journals in mathematics, and shaped it for two decades. He served on the Fields Medal committee in 1978 and 1994 and on the international jury of the Balzan Prize from 1985. Later honours included the Wolf Prize in 1993, shared with Mikhail Gromov, the German order Pour le Mérite for Sciences and Arts and the Légion d’honneur in 1995, and the Cantor Medal of the German Mathematical Society in 1996.

The Abel Prize and legacy, 2000–2021

After retiring from the Collège de France in 2000, Tits became the first holder of the Vallée-Poussin Chair at the University of Louvain, giving his inaugural lecture on 18 October 2001 on buildings as a geometric approach to simple algebraic groups and Kac–Moody groups. In 2002 he and Richard Weiss published Moufang Polygons, completing a classification he had pursued for three decades.

In 2008 the Norwegian Academy of Science and Letters awarded the Abel Prize jointly to Tits and John Griggs Thompson, “for their profound achievements in algebra and in particular for shaping modern group theory”. The committee stressed his geometric vision of groups: Thompson’s work on finite groups and Tits’s buildings had approached the classification of finite simple groups from two sides. In 2013 the European Mathematical Society published his collected works in four volumes.

Jacques Tits died in Paris on 5 December 2021, aged 91. A tribute by Jean-Pierre Bourguignon, Michel Broué, Alain Connes and Étienne Ghys appeared in Nature in January 2022.

Legacy

• Groups of Lie type: BN pairs and buildings give a uniform treatment of all simple algebraic groups and finite groups of Lie type, and the classification of finite simple groups uses them throughout.
• Number theory and representation theory: Bruhat–Tits buildings are the standard tool for \(p\)-adic groups, their representations and their arithmetic subgroups.
• Geometric group theory: the Tits alternative is the prototype of many dichotomies, and buildings are basic examples of CAT(0) spaces, used in rigidity theory (including Grigory Margulis’s superrigidity over local fields) and in the construction of exotic lattices.
• Combinatorics: generalized polygons and Coxeter groups are central in finite geometry and algebraic combinatorics.
• Names: Tits buildings, the Tits alternative, the Tits group, Bruhat–Tits theory, the Tits cone of a Coxeter group, Artin–Tits groups, the Kneser–Tits problem, the Freudenthal–Tits magic square and the Kantor–Koecher–Tits construction all carry his name.

Connections

People linked to Jacques Tits across this collection. A name without a link has no page here yet. See Jacques Tits in the network of all 337 lives →

Teachers and advisors

• Paul Libois 1950

Students

• Pierre Deligne
• Francis Buekenhout 1965
• Jens Carsten Jantzen
• Karl-Otto Stöhr

Influenced by

• Claude Chevalley
• H. S. M. Coxeter

Influenced

• Grigory Margulis

Colleagues and collaborators

• Grigory Margulis
• Jean-Pierre Serre
• Mikhail Gromov
• Émile Borel 1965
• François Bruhat 1972-1984
• Hans Freudenthal
• John Griggs Thompson
• Mark Ronan
• Richard Weiss 2002

Also mentioned in this life

• André Weil
• Emil Artin
• Georg Cantor
• Arthur Cayley
• Leonhard Euler
• Felix Klein
• Robert Langlands
• Sophus Lie
• John Milnor
• Bernhard Riemann
• John von Neumann
• Hermann Weyl
• Oscar Zariski

Fields

• group theory 9 lives

On CoursesHub

• Group Theory · Physical Mathematics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Jacques Tits, MacTutor History of Mathematics.
2. Jacques Tits, Wikipedia.
3. Collège de France, Jacques Tits: biography.
4. The Abel Prize, Jacques Tits, Abel Prize laureate, dies at 91; The Abel Committee’s citation (2008); popular account of the 2008 prize.
5. IHÉS, Tribute to Jacques Tits.
6. European Mathematical Society, Jacques Tits (1930–2021).
7. M. Raussen and C. Skau, Interview with John G. Thompson and Jacques Tits, Newsletter of the EMS, 2008.

Mathematics

1. J. Tits, Œuvres / Collected Works, 4 vols., EMS Press, 2013.
2. J. Tits, Sur les analogues algébriques des groupes semi-simples complexes, Colloque d’algèbre supérieure, Bruxelles, 1957.
3. J. Tits, Sur la trialité et certains groupes qui s’en déduisent, Publ. Math. IHÉS 2 (1959).
4. J. Tits, Algebraic and abstract simple groups, Ann. of Math. 80 (1964).
5. A. Borel and J. Tits, Groupes réductifs, Publ. Math. IHÉS 27 (1965).
6. J. Tits, Free subgroups in linear groups, J. Algebra 20 (1972).
7. F. Bruhat and J. Tits, Groupes réductifs sur un corps local I, II, Publ. Math. IHÉS 41 (1972), 60 (1984).
8. J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math. 386, Springer, 1974.
9. J. Tits, A local approach to buildings, in The Geometric Vein, Springer, 1981.
10. J. Tits, Uniqueness and presentation of Kac–Moody groups over fields, J. Algebra 105 (1987).
11. J. Tits and R. M. Weiss, Moufang Polygons, Springer, 2002.
12. P. Abramenko and K. S. Brown, Buildings: Theory and Applications, Springer, 2008.
13. J.-P. Serre, Trees, Springer, 1980.
14. M. Ronan, Lectures on Buildings, University of Chicago Press, updated edition 2009.
15. Building (mathematics) and Tits alternative, Wikipedia.