Moufang polygons.

*(English)*Zbl 1010.20017
Springer Monographs in Mathematics. Berlin: Springer. ix, 535 p. (2002).

This book contains the complete classification of all Moufang generalized polygons, including the full proof. This is a major result, which can only be put in the right perspective if one knows its peculiar history.

A generalized polygon is a rank 2 spherical building, or, alternatively, a bipartite graph of diameter \(n\geq 2\) and girth \(2n\) each vertex of which has valency at least 3 (and then we talk about generalized \(n\)-gons). For \(n=2\), this is just a complete bipartite graph and the corresponding rank 2 building is reducible. For \(n>2\), a Moufang \(n\)-gon is a generalized \(n\)-gon such that, for each path \((x_0,x_1,\dots,x_n)\) the automorphism group fixing all vertices adjacent to one of \(x_1,x_2,\dots,x_{n-1}\) acts transitively on the set of vertices adjacent to \(x_0\) but different from \(x_1\) (or, equivalently, on the set of vertices adjacent to \(x_n\) but different from \(x_{n-1}\)). This condition was introduced by Jacques Tits in the addendum of his famous book “Buildings of spherical type and finite BN-pairs” [Lect. Notes Math. 386 (1974; Zbl 0295.20047)]. The observation that led to this definition was that every irreducible spherical building of rank \(\geq 3\) satisfies a likewise condition, and that every irreducible residue of rank 2, which is a generalized polygon, is a Moufang polygon (Tits hereby generalized the notion of a Moufang projective plane – the case of \(n=3\) – that was introduced earlier by Pickert because of the fact that Ruth Moufang had characterized and studied these planes in a geometric way in the case there was no Fano sub-configuration). Moreover, this condition is satisfied in a natural way by all examples related to algebraic groups of relative rank 2. In fact, these examples even satisfy stronger assumptions arising from the underlying irreducible root systems (the Steinberg relations). It turned out that Tits could deduce these commutation relations from the Moufang condition (this was announced by Tits from the beginning, but the first explicit proof of that appeared in 1994 [Bull. Belg. Math. Soc. - Simon Stevin 1, No. 3, 455-468 (1994; Zbl 0811.51006)]). This fact convinced Tits that a full classification should be possible, and he started to think about it already in the late sixties. The rough conjecture was that all Moufang polygons are related to algebraic groups, classical groups or mixed groups. The case \(n=3\) was already done in the literature. The case \(n=6\) seemed vaguely related to the case \(n=3\) and it was that case that Tits readily solved before 1970. In the early seventies he dealt with the octagons (although the paper only appeared in 1983 [Am. J. Math. 105, 539-594 (1983; Zbl 0521.20016)]; there was never a paper on the hexagons but there were unpublished notes). For \(n=4\), there was a precise conjecture, and for all other cases (\(n\neq 3,4,6,8\)) Tits began to prove nonexistence in a first paper that appeared in 1976 [Invent. Math. 36, 275-284 (1976; Zbl 0369.20004)], finishing in a second one in 1979 [ibid. 51, 267-269 (1979; Zbl 0429.20029)]. At the same time, inspired by Tits’ first paper on the nonexistence, Richard Weiss put together a simpler and more general proof of that part, and this was published in the same issue of Invent. Math. as Tits’ second paper [ibid. 51, 261-266 (1979; Zbl 0409.05033)]. So, at that point, only the Moufang quadrangles remained unclassified. In 1996, Tits started to lecture on the classification of Moufang polygons with the aim to finally finish and write down the case \(n=4\). At the same time Richard Weiss was visiting Gent, and we both attended some lectures of Tits. Then Tits and Weiss decided to write down the classification in a book, and this book is being reviewed now.

The book uses original ideas of Tits mixed with new ideas of Weiss. In particular, the classification only uses elementary group theory and no knowledge of algebraic groups is needed, unlike the original approach of Tits.

One remarkable fact is that this approach revealed (in 1997) the existence of a new class of Moufang quadrangles not included in Tits’ original conjecture, and seemingly unrelated to algebraic groups, mixed groups or classical ones. Bernhard Mühlherr and the reviewer identified this new example as the fixed point structure of an involution in a mixed building of type \(F_4\), and hence Tits’ rough conjecture remains true.

The case \(n=4\) occupies more than half of the book, showing that this case is indeed the hardest. The general method in all cases runs as follows. First one uses the commutation relations to define a certain algebraic structure, and then one classifies the algebraic structures. This classification is carried out in full and in detail. There are much algebra and a lot of clever calculations involved, but the book is self contained.

In an appendix, the authors explain the relation with algebraic groups using the so-called Tits diagrams (which are called indices in the book). This is actually a very useful introduction into the theory and even into the classification of semisimple algebraic groups.

As mentioned before, the Moufang condition was introduced by Tits in the addendum of the monograph in which he classifies the irreducible spherical buildings of rank \(\geq 3\). The fact that this condition allows for a full classification in the rank 2 case may have led to the definition of Moufang polygons, but the motivation to write it in the above mentioned appendix must be found in the following. Tits was hoping that the classification of Moufang polygons would allow for an alternative and more elementary proof of the classification of irreducible spherical buildings of rank \(\geq 3\). This is indeed true, and such a proof is also contained in the book under review.

The book also contains a discussion of the full automorphism groups of the Moufang polygons. These groups are determined in all but three cases: a certain class of hexagons related to exceptional Jordan algebras, the class of exceptional Moufang quadrangles of types \(E_6\), \(E_7\), \(E_8\), and the new class of Moufang quadrangles (those of type \(F_4\)). The latter case is solved in the meantime by Tom De Medts in his doctoral thesis.

Let me also remark that the Moufang condition can be generalized to the non spherical case, and also these buildings can be classified (main results by Bernnard Mühlherr); the classification makes use of the classification of Moufang polygons, and has ramifications in the theory of Kac-Moody groups and their automorphism groups.

So, in conclusion, the book is a Bible for everyone interested in classification results related to spherical buildings. It is written in a very clear and concise way. It should be in the library of every mathematician as one of the major results in the theory of (Tits) buildings, (combinatorial) incidence geometry and (algebraic) group theory.

A generalized polygon is a rank 2 spherical building, or, alternatively, a bipartite graph of diameter \(n\geq 2\) and girth \(2n\) each vertex of which has valency at least 3 (and then we talk about generalized \(n\)-gons). For \(n=2\), this is just a complete bipartite graph and the corresponding rank 2 building is reducible. For \(n>2\), a Moufang \(n\)-gon is a generalized \(n\)-gon such that, for each path \((x_0,x_1,\dots,x_n)\) the automorphism group fixing all vertices adjacent to one of \(x_1,x_2,\dots,x_{n-1}\) acts transitively on the set of vertices adjacent to \(x_0\) but different from \(x_1\) (or, equivalently, on the set of vertices adjacent to \(x_n\) but different from \(x_{n-1}\)). This condition was introduced by Jacques Tits in the addendum of his famous book “Buildings of spherical type and finite BN-pairs” [Lect. Notes Math. 386 (1974; Zbl 0295.20047)]. The observation that led to this definition was that every irreducible spherical building of rank \(\geq 3\) satisfies a likewise condition, and that every irreducible residue of rank 2, which is a generalized polygon, is a Moufang polygon (Tits hereby generalized the notion of a Moufang projective plane – the case of \(n=3\) – that was introduced earlier by Pickert because of the fact that Ruth Moufang had characterized and studied these planes in a geometric way in the case there was no Fano sub-configuration). Moreover, this condition is satisfied in a natural way by all examples related to algebraic groups of relative rank 2. In fact, these examples even satisfy stronger assumptions arising from the underlying irreducible root systems (the Steinberg relations). It turned out that Tits could deduce these commutation relations from the Moufang condition (this was announced by Tits from the beginning, but the first explicit proof of that appeared in 1994 [Bull. Belg. Math. Soc. - Simon Stevin 1, No. 3, 455-468 (1994; Zbl 0811.51006)]). This fact convinced Tits that a full classification should be possible, and he started to think about it already in the late sixties. The rough conjecture was that all Moufang polygons are related to algebraic groups, classical groups or mixed groups. The case \(n=3\) was already done in the literature. The case \(n=6\) seemed vaguely related to the case \(n=3\) and it was that case that Tits readily solved before 1970. In the early seventies he dealt with the octagons (although the paper only appeared in 1983 [Am. J. Math. 105, 539-594 (1983; Zbl 0521.20016)]; there was never a paper on the hexagons but there were unpublished notes). For \(n=4\), there was a precise conjecture, and for all other cases (\(n\neq 3,4,6,8\)) Tits began to prove nonexistence in a first paper that appeared in 1976 [Invent. Math. 36, 275-284 (1976; Zbl 0369.20004)], finishing in a second one in 1979 [ibid. 51, 267-269 (1979; Zbl 0429.20029)]. At the same time, inspired by Tits’ first paper on the nonexistence, Richard Weiss put together a simpler and more general proof of that part, and this was published in the same issue of Invent. Math. as Tits’ second paper [ibid. 51, 261-266 (1979; Zbl 0409.05033)]. So, at that point, only the Moufang quadrangles remained unclassified. In 1996, Tits started to lecture on the classification of Moufang polygons with the aim to finally finish and write down the case \(n=4\). At the same time Richard Weiss was visiting Gent, and we both attended some lectures of Tits. Then Tits and Weiss decided to write down the classification in a book, and this book is being reviewed now.

The book uses original ideas of Tits mixed with new ideas of Weiss. In particular, the classification only uses elementary group theory and no knowledge of algebraic groups is needed, unlike the original approach of Tits.

One remarkable fact is that this approach revealed (in 1997) the existence of a new class of Moufang quadrangles not included in Tits’ original conjecture, and seemingly unrelated to algebraic groups, mixed groups or classical ones. Bernhard Mühlherr and the reviewer identified this new example as the fixed point structure of an involution in a mixed building of type \(F_4\), and hence Tits’ rough conjecture remains true.

The case \(n=4\) occupies more than half of the book, showing that this case is indeed the hardest. The general method in all cases runs as follows. First one uses the commutation relations to define a certain algebraic structure, and then one classifies the algebraic structures. This classification is carried out in full and in detail. There are much algebra and a lot of clever calculations involved, but the book is self contained.

In an appendix, the authors explain the relation with algebraic groups using the so-called Tits diagrams (which are called indices in the book). This is actually a very useful introduction into the theory and even into the classification of semisimple algebraic groups.

As mentioned before, the Moufang condition was introduced by Tits in the addendum of the monograph in which he classifies the irreducible spherical buildings of rank \(\geq 3\). The fact that this condition allows for a full classification in the rank 2 case may have led to the definition of Moufang polygons, but the motivation to write it in the above mentioned appendix must be found in the following. Tits was hoping that the classification of Moufang polygons would allow for an alternative and more elementary proof of the classification of irreducible spherical buildings of rank \(\geq 3\). This is indeed true, and such a proof is also contained in the book under review.

The book also contains a discussion of the full automorphism groups of the Moufang polygons. These groups are determined in all but three cases: a certain class of hexagons related to exceptional Jordan algebras, the class of exceptional Moufang quadrangles of types \(E_6\), \(E_7\), \(E_8\), and the new class of Moufang quadrangles (those of type \(F_4\)). The latter case is solved in the meantime by Tom De Medts in his doctoral thesis.

Let me also remark that the Moufang condition can be generalized to the non spherical case, and also these buildings can be classified (main results by Bernnard Mühlherr); the classification makes use of the classification of Moufang polygons, and has ramifications in the theory of Kac-Moody groups and their automorphism groups.

So, in conclusion, the book is a Bible for everyone interested in classification results related to spherical buildings. It is written in a very clear and concise way. It should be in the library of every mathematician as one of the major results in the theory of (Tits) buildings, (combinatorial) incidence geometry and (algebraic) group theory.

Reviewer: H.Van Maldeghem (Gent)

##### MSC:

20E42 | Groups with a \(BN\)-pair; buildings |

51E24 | Buildings and the geometry of diagrams |

51E12 | Generalized quadrangles and generalized polygons in finite geometry |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

51-02 | Research exposition (monographs, survey articles) pertaining to geometry |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

20G15 | Linear algebraic groups over arbitrary fields |