Œuvres. Collected works. Vol. I--IV. Edited by Francis Buekenhout, Bernhard Matthias Mühlherr, Jean-Pierre Tignol, Hendrik Van Maldeghem (Q382120)

Jacques Tits is one of the outstanding mathematicians of our time. His published (and unpublished) works, most of which are collected in these impressive four volumes, encompass a wide area of modern mathematics. The central and recurring topics in his œuvre are certainly group theory and geometry. However, group theory itself is a vast area. DuringNEWLINEthe last six decades, Jacques Tits has contributed important results to finite group theory, to Lie theory, to algebraic group theory, to combinatorial and geometric group theory, to Kac-Moody theory, and to arithmetic geometry. Probably he is one of the few present-day mathematicians who may claim to have an overview of group theory in its entirety. In some of theseNEWLINEareas, he was among those who laid the foundations, while in others, he pushed existing results much further. The task to describe what is in these four volumes is thus somewhat intimidating. I will try to highlight and describe some of their contents, led partially by my own mathematical interests and knowledge. The four volumes group his work, of course, in a chronological order. In what follows, I will try to group some of Tits' contributions by their topics.NEWLINENEWLINENEWLINE\textbf{Multiply transitive groups and Lie groups}NEWLINENEWLINEA group \(G\) acting on a set \(X\) is called \(k\)-transitive if, given any two \(k\)-tuples of distinct elements of \(X\), there is a group element \(g\) that maps one \(k\)-tuple to the other. (We assume here that \(X\) contains at least \(k\) elements.) The action is called sharply \(k\)-transitive if the group element in question is uniquely determined. The reader should be warned here that the terminology has changed over the years, and also varies in Tits' articles -- what we call sharply \(k\)-transitive here was called earlier \(k\)-transitive.NEWLINENEWLINEIn any case, the symmetric group on \(k\) letters is an example of a sharply \(k\)-transitive group. At the same time, its action is \(\ell\)-transitive for all \(\ell\leq k\). If \(F\) is a field, then the projective linear group \(\mathrm{PGL}_2(F)\) acts sharply \(3\)-transitively on the projective line \(P=F\cup\{\infty\}\). Surprisingly, there exist no infinite sharply \(k\)-transitive groups for \(k\geq4\). Two of the sporadic simple groups, the Mathieu groups \(M_{11}\) and \(M_{12}\), are among the finite sharply \(4\)- and \(5\)-transitive groups. Tits' PhD ThesisNEWLINE[Généralisations des groupes projectifs basées sur leurs propriétés de transitivité. Acad. Roy. Belgique, Cl. Sci., Mém., Coll. 8, 27, No. 2 (1952; Zbl 0048.25702)] discusses these transitivity properties, first in the case of the projective linear groups, and then in general. It contains, in particular, the classification of all sharply \(k\)-transitive groups with \(k\geq 4\), a remarkable result.NEWLINENEWLINEIn the long article [``Sur certaines classes d'espaces homogènes de groupes de Lie'', ibid. 8, 29, No. 3 (1955; Zbl 0067.12301)] (roughly equivalent to a \textit{Habilitationsschrift}),NEWLINETits obtains a large number of interesting results about homogeneous spaces of Lie groups and locally compact groups, based on the transitivity properties of the actions. My favorite is the classification of \(2\)-transitive Lie groups. Suppose that a Lie group \(G\) acts faithfully (i.e., with trivial kernel) and \(2\)-transitively on a connected manifold \(X\). In present-dayNEWLINEterminology, he proves that there are precisely three cases. (1)~The identity component \(G^\circ\) of \(G\) is the projective special linear group over the real numbers, the complex numbers, the quaternions or the Cayley division algebra, acting on the corresponding projective space. (2)~\(G^\circ\) is the connected isometry group of a hyperbolic space over the real numbers, the complex numbers, the quaternions or the Cayley division algebra, acting on its boundary. (3)~\(G=H\ltimes\mathbb R^m\), where \(H\subseteq \mathrm{GL}_m\mathbb R\) is a closed linear Lie group acting transitively on the nonzero vectors of \(\mathbb R^m\). In particular, the topological space \(X\) is either a projective space or a sphere of a real vector space. Related results on the topology of \(X\) had been obtained at the same time, and independently, by A.~Borel. However, Borel did not classify the corresponding Lie groups.NEWLINENEWLINEIn the same article, Tits obtains many more equally interesting classification results. For example, he classifies transitive faithful \(G\)-actions on manifolds where every point stabilizer \(G_x\) acts transitively on the tangent directions at \(x\). The results in [Zbl 0067.12301] seem to be less known than they deserve. Some particular cases were re-discovered and reproved several times by others. This is probably also due to the fact that prior to the publication of Tits' collected works, it was difficult to get hold of a copy of [Zbl 0067.12301].NEWLINENEWLINENEWLINE\textbf{The structure of Lie groups and algebraic groups}NEWLINENEWLINEThe article [Zbl 0067.12301] depends very much on the structure theory of semisimple LieNEWLINEgroups. As Tits himself pointed out, the work on transitivity properties ofNEWLINELie groups led him to the combinatorial structure of parabolic subgroups,NEWLINEand finally to the definition of buildings. Also, Dynkin diagramsNEWLINE(called ``figure de Schläfli'') of simple Lie groups appear in [Zbl 0067.12301], withNEWLINEa reference to Dynkin's work. In the early 60s, Tits wrote a whole seriesNEWLINEof papers on real forms of the exceptional Lie groups \(E_6\), \(E_7\) and \(E_8\).NEWLINEIn these articles, his focus shifts already from Lie groups to semisimpleNEWLINEalgebraic groups over arbitrary fields. At the same time the associatedNEWLINEcombinatorial structures emerge, which led him subsequently to the discoveryNEWLINEof buildings.NEWLINENEWLINENEWLINENEWLINEI also want to mention the short Lecture Notes volumeNEWLINE[Tabellen zu den einfachen Lie-Gruppen und ihren Darstellungen.NEWLINELecture Notes in Mathematics 40. Berlin-Heidelberg-New York:NEWLINESpringer-Verlag (1967; Zbl 0166.29703)]NEWLINEon the representation theory of Lie groups.NEWLINEIn combination withNEWLINE[``Représentations linéaires irréductibles d'un groupeNEWLINEréductif sur un corps quelconque'', J. Reine Angew. Math. 247, 196--220 (1971; Zbl 0227.20015)],NEWLINEwhere rational representations of reductive groupsNEWLINEover arbitrary fields (and in particular of real Lie groups) areNEWLINEclassified, theseNEWLINEtables are indispensable for any Lie theorist.NEWLINENEWLINENEWLINENEWLINEHis joint paper with \textit{A.~Borel}NEWLINE[``Groupes réductifs'', Publ. Math., Inst. Hautes Étud. Sci. 27,NEWLINE659--755 (1965; Zbl 0145.17402)]NEWLINEon the structure of reductive algebraic groups overNEWLINEarbitrary fields still stands as a landmark paper and as a standard reference.NEWLINEThe classification and the structure theory of semisimple algebraic groupsNEWLINEover arbitrary fields is a recurring topic in Tits' work. His BoulderNEWLINEarticleNEWLINE[``Classification of algebraic semi-simple groups'',NEWLINEProc. Symp. Pure Math. 9, 33--62 (1966; Zbl 0238.20052)]NEWLINEsummarizes his results in a very nice and reader-friendly way.NEWLINEHistory and terminology are not always fair: what is nowadays often calledNEWLINEthe ``Satake diagram'' of a semisimple algebraic group should properlyNEWLINEbe called the ``Tits diagram''. These decorated Dynkin diagrams, togetherNEWLINEwith the datum of the so-called anisotropic kernel, classify algebraic groupsNEWLINEup to isogeny over arbitrary fields.NEWLINEThis result is certainly one of the major mathematical achievements ofNEWLINEJacques Tits.NEWLINENEWLINENEWLINENEWLINEAnother recurring theme in Tits' work is the underlying ``abstract'' groupNEWLINEstructure of a simple algebraic group or Lie group. InNEWLINE[``Algebraic and abstract simple groups'',NEWLINEAnn. Math. (2) 80, 313--329 (1964; Zbl 0131.26501)],NEWLINEhe shows that if \(G\) is an absolutelyNEWLINEsimple algebraic group, defined over a sufficiently large field \(F\),NEWLINEand if \(G^\dagger\subseteq G(F)\) is the subgroup generated by theNEWLINE\(F\)-rational unipotent elements, then \(G^\dagger/\mathrm{Cen}(G^\dagger)\) isNEWLINEsimple as an abstract group. His proof uses spherical buildings andNEWLINE\((B,N)\)-pairsNEWLINEin a systematic way. In another joint paper with \textit{A.~Borel}NEWLINE[``Homomorphismes `abstraits' de groupes algébriques simples'',NEWLINEAnn. Math. (2) 97, 499--571 (1973; Zbl 0272.14013)],NEWLINEhe provesNEWLINE(essentially) that if the groups of rational points of two absolutely simpleNEWLINEisotropic algebraic groups \(G\) and \(G'\), possibly defined over differentNEWLINEfields \(F\) and \(F'\), are isomorphic as abstract groups, then \(G\) and \(G'\) areNEWLINEisomorphic as algebraic groups. (The precise statement is somewhat moreNEWLINEelaborate, but this is basically the result.) Tits needed this forNEWLINEhis work on the automorphism groups of spherical buildings associatedNEWLINEto semisimple algebraic groups inNEWLINE[Buildings of spherical type and finite BN-pairs.NEWLINELecture Notes in Mathematics 386. Berlin-Heidelberg-New York: Springer-VerlagNEWLINE(1974; Zbl 0295.20047)]. It would be interesting to seeNEWLINEif nowadays,NEWLINEwith the advanced structure theory of buildings at hand, buildingsNEWLINEcould be used to prove this deep result in a completely geometric way.NEWLINENEWLINENEWLINENEWLINE\textbf{Buildings}NEWLINENEWLINENEWLINENEWLINEThe discovery of the various kinds of buildings is probably the mathematicalNEWLINEtopic which is associated most often with Jacques Tits.NEWLINEIt had been known at least since the 19th century that the projectiveNEWLINEgeometry, that is, the collection of all nontrivial subspaces of a givenNEWLINEvector space \(V\), is an extremely useful tool for studying theNEWLINEgeneral linear group \(\mathrm{GL}(V)\). It was also known that this projectiveNEWLINEgeometry can be described by a relatively short list of geometric incidenceNEWLINEaxioms. The fundamental theorem of projectiveNEWLINEgeometry asserts that (1)~every projective geometry of rank atNEWLINEleast \(3\) (in the synthetic sense) arises from some vector space overNEWLINEsome field, and that (2)~every automorphism of this geometryNEWLINEcomes from a semi-linear automorphism of the vector space.NEWLINEThese are extremely useful facts, as they guarantee, in theNEWLINEpresence of very few combinatorial axioms, the existence ofNEWLINEa field, a vector space, and a linear group. Similar constructionsNEWLINEwere known for the other classical groups, like symplectic groupsNEWLINEor orthogonal groups. However, it was not so clear which combinatorialNEWLINEstructures should correspond to the exceptional groups likeNEWLINE\(E_6\), \(E_7\) and \(E_8\). Geometers like H. Freudenthal devoted a largeNEWLINEnumber of papers to this question. However, the combinatorialNEWLINEconstructions looked complicated and a unifying theme was missing.NEWLINENEWLINENEWLINENEWLINEThis changed completely with Tits' discoveries. Firstly, he showedNEWLINEin a uniform way that to every group \(G\) with a Tits system orNEWLINE\((B,N)\)-pair, that is, with two subgroups \(B,N\subseteq G\) satisfyingNEWLINEa list of four axioms, one can associate a combinatorial structure,NEWLINEthe building \(\Delta(G)\). The building is a simplicial complex,NEWLINEand \(G\) acts simplicially on it. Secondly, the building itselfNEWLINEcan be characterized by a very short set of combinatorialNEWLINEaxioms. There is an associated Weyl group \(W=N/B\cap N\)NEWLINEthat describes the so-called type ofNEWLINEthe building. If \(W\) is finite, then the building is calledNEWLINEspherical. Tits' discovery was that one can find such aNEWLINE\((B,N)\)-pair in every simple isotropic \(F\)-algebraic group \(G\).NEWLINEIf the \(F\)-rank of \(G\) is at least~\(2\), thenNEWLINE\(G(F)\rtimes\mathrm{Aut}(F)\) is (essentially)NEWLINEthe group of all combinatorial automorphisms of the building \(\Delta\).NEWLINEThe \(G(F)\)-action on the building can then be used in orderNEWLINEto study \(G\) and its subgroups. This result is proved in Tits'NEWLINELecture Note [Zbl 0295.20047]. The other important result proved in [Zbl 0295.20047]NEWLINEis the classification of spherical buildings of rank at least~\(3\).NEWLINEThese spherical buildings arise from algebraicNEWLINEdata (such as fields, division algebras, Cayley algebras, quadratic forms).NEWLINEIn fact, almost all of them arise from simple \(F\)-algebraicNEWLINEgroups. The fundamental theorem of projective geometryNEWLINEis a special case of this result.NEWLINENEWLINENEWLINENEWLINEThis classification result from [Zbl 0295.20047], which one might call the fundamentalNEWLINEtheorem of spherical buildings, is the reason why buildings turnedNEWLINEout to be so useful in group theory (including the classification ofNEWLINEthe finite simple groups) and geometry. Again, the presenceNEWLINEof a combinatorial structure \(\Delta\) satisfying a short list of simple axiomsNEWLINEguarantees the presence of a field \(F\) and (in most cases) a simpleNEWLINE\(F\)-algebraic group acting on \(\Delta\). This had, for example, strikingNEWLINEapplications in Riemannian geometry in connection with manifolds ofNEWLINEnonpositive curvature and with isoparametric foliations.NEWLINENEWLINENEWLINENEWLINEAt the very end of [Zbl 0295.20047], the Moufang condition appears already. TheNEWLINEMoufang condition, a transitivity property of the automorphismNEWLINEgroup of the building, turned out to be an essential propertyNEWLINEof spherical buildings. In [Zbl 0295.20047] Tits shows that every sphericalNEWLINEbuilding of rank at least~\(3\) satisfies the Moufang condition.NEWLINEThe proof, albeit long, is strictly combinatorial. Once thisNEWLINEis shown, the algebra of root groups and commutator relationsNEWLINEmay be used to translate the classification into an algebraicNEWLINEproblem. To this end, Tits studies and classifies certainNEWLINEMoufang polygons, that is, spherical buildings ofNEWLINErank~\(2\) satisfying the Moufang condition, in several of his papers.NEWLINEThe complete classification was carried out by Tits and \textit{R. M. Weiss}NEWLINEin the monographNEWLINE[Moufang polygons. Berlin:NEWLINESpringer (2002; Zbl 1010.20017)], NEWLINEwhich is an almost mandatory supplement to Tits'NEWLINEcollected works.NEWLINENEWLINENEWLINENEWLINEThere is one other paper on buildingsNEWLINE[``A local approach to buildings'', in:NEWLINEThe geometric vein, The Coxeter Festschr., 519--547 (1982; ZblNEWLINE0496.51001)] that I want to mention here,NEWLINEwhere the ``local approach''NEWLINEto buildings is carried out. The paper is important for severalNEWLINEreasons. Firstly, it introduces a completely different approach to buildingsNEWLINEas chamber systems, that is, as generalized Cayley graphs.NEWLINEThis approach to buildings (which is shown to be functorially equivalentNEWLINEto the older one) is particularly well-suited for group-theoretic questions,NEWLINEwhile the older simplicial approach fits better into the worldNEWLINEof metric geometry and nonpositive curvature.NEWLINESecondly, this article gives a criterion when a simplicial complexNEWLINEthat looks locally like a building is actually a building orNEWLINEa quotient of a building. This result has become increasinglyNEWLINEimportant during the last years, for example in RiemannianNEWLINEgeometry. It has also led to the construction of exoticNEWLINEbuildings and lattices, starting with a finite complex ofNEWLINEfinite groups. This article goes well withNEWLINEthe fundamental papers by M.~Gromov on geometric group theory thatNEWLINEappeared at about the same time.NEWLINENEWLINENEWLINENEWLINE\textbf{Reductive algebraic groups over local fields}NEWLINENEWLINENEWLINENEWLINEIf \(G\) is a reductive isotropic group over a local field \(F\), thenNEWLINEthere is the Tits system \((B,N)\) and its spherical building \(\Delta\), asNEWLINEdescribed in the previous section. However, the valuation ofNEWLINE\(F\) gives rise to a richer subgroup structure of \(G(F)\)NEWLINEand eventually to a second Tits system \((I,N)\), with the same subgroupNEWLINE\(N\) but a different group \(I\subseteq G(F)\), the Iwahori subgroup.NEWLINEThe second building, \(X\), associated to \((I,N)\) is not of sphericalNEWLINEtype. Its Weyl group \(\overline W=N/N\cap I\) is an infinite EuclideanNEWLINEreflection group, and \(X\) is a Euclidean building.NEWLINEThe simplicial complex \(X\) is unbounded and carries aNEWLINEmetric of nonpositive curvature. There is a close relation betweenNEWLINEthe two buildings: the spherical building \(\Delta\) appears as theNEWLINE``boundary at infinity'' of the Euclidean building \(X\).NEWLINEIn a series of joint papers with \textit{F.~Bruhat}NEWLINE[``Groupes réductifs sur un corps local. I. Donnés radiciellesNEWLINEvaluées'',NEWLINEPubl. Math., Inst. Hautes Étud. Sci. 41, 5--251 (1972; ZblNEWLINE0254.14017); NEWLINE``Groupes réductifs sur un corps local. II. Schémas en groupes.NEWLINEExistence d'une donnée radicielle valuée'',NEWLINEibid. 60, 1--194 (1984; ZblNEWLINE0597.14041); NEWLINE``Schémas en groupes et immeubles des groupes classiques sur un corpsNEWLINElocal'', Bull. Soc. Math. Fr. 112, 259--301 (1984; Zbl 0565.14028)],NEWLINETits developed theNEWLINEstructure theory both of Euclidean buildings and of reductiveNEWLINEgroups over local fields. InNEWLINE[``Immeubles de type affine'', NEWLINELect. Notes Math. 1181, 159--190 (1986; Zbl 0611.20026)],NEWLINEhe outlined how the classificationNEWLINEof spherical buildings can be used to achieve a classification ofNEWLINEEuclidean buildings of dimension at least~\(3\). In this article, heNEWLINEconsiders in fact a much wider class of metric spaces of nonpositiveNEWLINEcurvature, which need not be buildings in the combinatorial sense.NEWLINENowadays, it is common to call these spaces nondiscrete Euclidean buildings.NEWLINESimilarly to \(\mathbb R\)-trees, these spaces may branch everywhere.NEWLINENEWLINENEWLINENEWLINEEuclidean buildings are the counterparts in arithmetic geometryNEWLINEof the Riemannian symmetric spaces of noncompact type.NEWLINEThey found numerous applications in various fields, and notablyNEWLINEprovide important toolsNEWLINEfor studying or constructing representationsNEWLINEof reductive groups, for the Langlands program, and forNEWLINEShimura varieties. Yet, they are also important in RiemannianNEWLINEgeometry and geometric group theory, where the (wide open)NEWLINErank rigidity problem predicts that Euclidean buildings andNEWLINEsymmetric spaces are basically the only irreducible locally compactNEWLINEhigher-rank spaces ofNEWLINEnonpositive curvature admitting a cocompact group action.NEWLINENEWLINENEWLINENEWLINE\textbf{Kac-Moody groups and twin buildings}NEWLINENEWLINENEWLINENEWLINEKac-Moody algebras, which are infinite-dimensional generalizationsNEWLINEof semi\-simple Lie algebras, were discovered independently byNEWLINEV.~Kac and R.~Moody. These algebras attracted immediately the attention ofNEWLINEtheoretical physicists, and they play nowadays a role in variousNEWLINEphysical models. The construction of Kac-Moody groups, however,NEWLINEturned out to be more complicated. For complex affine Kac-Moody algebrasNEWLINEone can construct corresponding groups as loop groups of compact Lie groups.NEWLINEHowever, this method fails for other algebras. InNEWLINE[``Groups and group functors attached to Kac-Moody data'',NEWLINELect. Notes Math. 1111, 193--223 (1985; Zbl 0572.17010); NEWLINE``Uniqueness and presentation of Kac-Moody groups over fields'',NEWLINEJ. Algebra 105, 542--573 (1987; Zbl 0626.22013); NEWLINE``Groupes associés aux algèbres de Kac-Moody'',NEWLINESémin. Bourbaki, Vol. 31, 41e année (1988/1989), Exp. No. 700,NEWLINEAstérisque 177--178, 7--31 (1989; Zbl 0705.17018); NEWLINE``Twin buildings and groups of Kac-Moody type'', in: NEWLINEM. Liebeck (ed.) et al., Groups, combinatorics andNEWLINEgeometry. Proceedings of the L.M.S. Durham symposium. Cambridge:NEWLINECambridge UniversityNEWLINEPress. Lond. Math. Soc. Lect. Note Ser. 165, 249--286 (1992; ZblNEWLINE0851.22023)],NEWLINETits shows that there is a group functor that associates to everyNEWLINEKac-Moody datum and every field \(F\) a split Kac-Moody group \(G(F)\).NEWLINEThe construction resembles the construction of complex simpleNEWLINELie groups in terms of the Steinberg relations. Associated toNEWLINE\(G\), there are now two Tits systems \((B_\pm,N)\) and twoNEWLINEbuildings \(\Delta_\pm\), sharing the same Weyl group \(W\).NEWLINEThe group \(G(F)\) acts on both buildings, preserving a mapNEWLINE\(\delta^*:\Delta^+\times\Delta^-\to W\), the codistanceNEWLINEfunction. This structure is called a twin building.NEWLINETwin buildings over finite fields have been used, for example, byNEWLINEB.~Rémy to construct new simple locally compact groupsNEWLINEand lattices acting on spaces of nonpositive curvature.NEWLINENEWLINENEWLINENEWLINE\textbf{Coxeter groups, the Tits alternative, trees, and the Monster}NEWLINENEWLINENEWLINENEWLINEThe geometry and structure of Coxeter groups is essential forNEWLINEunderstanding buildings. Tits has devoted several papers toNEWLINECoxeter groups, to their structure and to the solution of the word problem.NEWLINEBourbaki's volume [Groupes et algèbres de Lie. Chapitres IV, V et~VI.NEWLINEActualités Scientifiques et Industrielles 1337. Paris: Hermann \&NEWLINECie. (1968; Zbl 0186.33001)] wasNEWLINEheavily influenced by Tits. These articles are stillNEWLINEthe standard references for Coxeter groups and root systems.NEWLINENEWLINENEWLINENEWLINESuch questions belong to the wider area of geometric group theory.NEWLINEPerhaps the most famous contribution of Tits here is whatNEWLINEis nowadays called the Tits alternativeNEWLINE[``Free subgroups in linear groups'',NEWLINEJ. Algebra 20, 250--270 (1972; Zbl 0236.20032)]:NEWLINEIf \(\Gamma\) is aNEWLINEfinitely generated linear group, then \(\Gamma\) either isNEWLINEvirtually solvable or it contains the free groupNEWLINEon two generators.NEWLINENEWLINENEWLINENEWLINETrees are special cases of Euclidean buildings. Tits studiesNEWLINEgroup actions on trees and related automorphism groups inNEWLINE[``Sur le groupe des automorphismes d'un arbre'',NEWLINEin: Essays Topol. Relat. Top., Mém. dédiés à Georges de Rham, 188--211NEWLINE(1970; Zbl 0214.51301); NEWLINE``A `theorem of Lie-Kolchin' for trees'',NEWLINEin: Contrib. to Algebra, Collect. Pap. dedic. E. Kolchin, 377--388 (1977;NEWLINEZbl 0373.20039)].NEWLINEHe proves a simplicity criterion for groups acting on treesNEWLINEin [Zbl 0214.51301]. In [Zbl 0373.20039] he introduces what is nowadays called an \(\mathbb R\)-tree,NEWLINEand he proves that a solvable group acting without fixed pointNEWLINEon such a tree fixes one end or two ends.NEWLINE\(\mathbb R\)-trees have since then become quite important inNEWLINEgeometric group theory and in Riemannian geometry, for example in connectionNEWLINEwith degenerations of hyperbolic structures on closed manifolds.NEWLINENEWLINENEWLINENEWLINETits' work had also a significant impact on finite group theory.NEWLINEThe classification of the finite spherical buildings of rankNEWLINEat least~\(3\), carried out in [Zbl 0295.20047], plays an important role in theNEWLINEclassification of the finite simple groups. Tits investigatedNEWLINEalso the sporadic finite simple groups, and the Tits group of orderNEWLINE\(2^{11}3^35^213 \) was discovered by him [Zbl 0131.26501]. He wrote in particularNEWLINEtwo papers on the Monster, simplifying Griess' constructionNEWLINE[``On R. Griess' `friendly giant''',NEWLINEInvent. Math. 78, 491--499 (1984; Zbl 0548.20011); NEWLINE``Le monstre (d'après R. Griess, B. Fischer et al.)'',NEWLINESémin. Bourbaki, 36ème année, Vol. 1983/84, Exp. No. 620, AstérisqueNEWLINE121--122, 105--122 (1985; Zbl 0548.20010)]NEWLINEand one on the moonshine moduleNEWLINE[``Le module du `moonshine' (d'après I. Frenkel, J. Lepowsky etNEWLINEA. Meurman)'',NEWLINESémin. Bourbaki, 39ème année, Vol. 1986/87, Exp. No. 684, AstérisqueNEWLINE152/153, 285--303 (1987; Zbl 0699.20015)].NEWLINENEWLINENEWLINENEWLINE\textbf{The volumes}NEWLINENEWLINENEWLINENEWLINEJacques Tits' works appears here in four volumes, published by the EuropeanNEWLINEMathematics SocietyNEWLINEand edited by F. Buekenhout, B. Mühlherr, J.-P. Tignol andNEWLINEH. Van Maldeghem. The editors have done a great job collectingNEWLINEand reproducing Tits' works. This was apparently not alwaysNEWLINEan easy task, as there are several papers that were difficult toNEWLINEobtain (and to reproduce).NEWLINEAlso, they included some manuscripts which were previouslyNEWLINEunpublished. In particular, the editors have included (and re-typed)NEWLINEtwo important sets of lecture notes from YaleNEWLINE[Lectures on algebraic groups (1967), [B1];NEWLINEAffine buildings, arithmetic groups and finite geometries (1984), [B2]].NEWLINEEach volume closes with an interesting list of comments andNEWLINEnotes by the editors, pointing towards further developments,NEWLINEcorrections and complements. EveryoneNEWLINEinterested in group theory should have access to theseNEWLINEfour volumes. What is not included are the \textit{RésumésNEWLINEde cours} articles with Tits' contributions from 1973 to 2000.NEWLINEBut theseNEWLINEare available from the Société Mathématique de FranceNEWLINEin a nice separate volumeNEWLINE[\textit{J.~Tits}, Résumés des cours au Collège de France (1973--2000).NEWLINEDocuments Mathématiques (Paris) 12. Paris: SociétéNEWLINEMathématique de France (2013; Zbl 1286.01001)].NEWLINEAnd finally, the monograph on Moufang polygons by Tits and WeissNEWLINE[Zbl 1010.20017] NEWLINEshould be mentioned here. The combination of these six booksNEWLINEencompasses some of the finest group theory and geometry of our time.