Groupes stables. Une tentative de conciliation entre la Géométrie Algébrique et la Logique Mathématique.

*(French)*Zbl 0633.03019
Nur Al-Mantiq Wal-Ma’rifah, 2. Villeurbanne, France: Bruno Poizat. VI, 215 p.; FF 100.00; $ 18.00 (1987).

This is a beautiful book in which almost everything known about stable groups appears. The author stresses the connections with algebraic geometry and algebraic groups. He is also concerned with exactly what properties are required to develop the theory of groups of finite Morley rank. In particular he shows that groups of finite Morley rank are exactly the groups studied by the Soviet mathematician Borovik.

In Chapter 1, elementary properties of stable groups are considered: chain conditions, connected components etc. Also the Baur-Cherlin- Macintyre theorem is proved: an \(\aleph_ 0\)-categorical stable group is nilpotent-by-finite, an \(\aleph_ 0\)-categorical \(\omega\)-stable group is abelian-by-finite.

In Chapter 2, the machinery for the study of groups of finite Morley rank is set up. Basic properties of \(\omega\)-stable groups (generics etc.) are also proved. Zilber’s indecomposability theorem, and Lascar’s theorem on groups of finite Morley rank (such groups are “finite-dimensional”, their dimensions are given by the quotients in a definable composition series) are proved. Also so-called binding groups are introduced: if there is some “strong connection” between definable sets X and Y in an (\(\omega\)-stable) structure M then the action of \(Aut_ X(M)\) of Y is definable; so we obtain a definable group. A consequence is that a nonalmost strongly minimal \(\aleph_ 1\)-categorical structure interprets an infinite group.

Chapter 3, perhaps the nicest chapter of the book, is concerned with the algebraic structure of groups of finite Morley rank. The point of view is Cherlin’s conjecture: Groups of finite Morley rank are “basically” algebraic groups (over an algebraically closed field). (“Basically” means except for trivial counterexamples.) Poizat isolates an obstruction to this program, the possible existence of bad groups; nonsolvable connected groups of finite Morley rank all of whose proper definable subgroups are nilpotent-by-finite. Cherlin had already shown that a simple nonbad group of Morley rank 3 is algebraic \((PSL_ 2(K))\). Poizat shows that bad groups can also give rise to bad fields (a field of finite Morley rank which is not a pure field). After proving Macintyre’s theorem that an \(\omega\)-stable field is algebraically closed, Poizat shows how a definable field comes from a definable action of an abelian group on another abelian group, all within a context of finite Morley rank. Pushing the argument a little more yields Nesin’s theorem: the derived group of a solvable connected group of finite Morley rank is nilpotent. Poizat shows how his method of proof yields the classical Lie-Kolchin theorem. Cherlin’s results on groups of rank 2, 3 are proved by means of studying the definable action of a connected group G on a strongly minimal set X. The possibilities were classified by Hrushovski; the rank of G is \(\leq 3\). If \(rank=2\), G is AGL(2,K) acting on the affine line, if \(rank=3\), G is PGL(2,K) acting on the projective line. Finally strange properties of bad groups are deduced, following Nesin.

Chapter 4 introduces the language of algebraic geometry and the theory of linear algebraic groups, showing how 1950’s algebraic geometry was really 1980’s model theory. Nice proofs are given of the elementary results: an affine algebraic group is linear, a centreless algebraic group is linear. Some space is devoted to proving that a group definable in a (pure) algebraically closed field K is definably isomorphic to an algebraic group over K. This theorem is attributed to Weil-Hrushovski, although should be to van den Dries (at least in characteristic 0). Finally the so-called Borel-Tits theorem is proved. This is rather a misnomer, the Borel-Tits theorem being concerned with isomorphism between rational points of algebraic groups. The new contribution here is rather that the geometric structure of a simple algebraic group over an algebraically closed field is first order definable in the abstract group structure.

Chapter 5 deals with the “pure” theory of stable groups, a theory basically due to Poizat himself. A new and elegant definition of generic types is given. Various results of Hrushovski are proved: an infinitely definable group (living in a stable structure) is enveloped by a definable group; a generically given group is generically isomorphic to an infinitely definable group. (The latter “generalises” a theorem of Andre Weil); a unidimensional stable theory is superstable (no group is mentioned in the hypothesis or conclusion, but binding groups play an essential role in the proof).

In Chapter 6, the Berline-Lascar theory of superstable groups is expounded; a simple superstable group has monomial U rank; the \(\alpha\)- indecomposability theorem, etc.

Chapter 7 is called “Poids” and is devoted largely to work of Hrushovski. The finite rank and superstable theory are simultaneously generalized by the theory of local weight, with respect to some regular type p. A “p-internal, p-connected” group has a finite “p-weight”. Anything one can prove about groups of finite Morley rank goes over to p- internal p-connected groups, replacing rank everywhere by ‘p-weight’ (Hrushovski more generally considers p-simple and p-semiregular groups). Moreover, by another result of Hrushovski a simple group or a field whose generic is nonorthogonal to a (regular) type p is p-internal and p- connected. Thus for example one sees that a stable field whose generic is nonorthogonal to a regular type is algebraically closed.

The illustrations “gentiment porno” which are interspersed throughout the book are somewhat out of place. For this kind of book they should be a little harder.

In Chapter 1, elementary properties of stable groups are considered: chain conditions, connected components etc. Also the Baur-Cherlin- Macintyre theorem is proved: an \(\aleph_ 0\)-categorical stable group is nilpotent-by-finite, an \(\aleph_ 0\)-categorical \(\omega\)-stable group is abelian-by-finite.

In Chapter 2, the machinery for the study of groups of finite Morley rank is set up. Basic properties of \(\omega\)-stable groups (generics etc.) are also proved. Zilber’s indecomposability theorem, and Lascar’s theorem on groups of finite Morley rank (such groups are “finite-dimensional”, their dimensions are given by the quotients in a definable composition series) are proved. Also so-called binding groups are introduced: if there is some “strong connection” between definable sets X and Y in an (\(\omega\)-stable) structure M then the action of \(Aut_ X(M)\) of Y is definable; so we obtain a definable group. A consequence is that a nonalmost strongly minimal \(\aleph_ 1\)-categorical structure interprets an infinite group.

Chapter 3, perhaps the nicest chapter of the book, is concerned with the algebraic structure of groups of finite Morley rank. The point of view is Cherlin’s conjecture: Groups of finite Morley rank are “basically” algebraic groups (over an algebraically closed field). (“Basically” means except for trivial counterexamples.) Poizat isolates an obstruction to this program, the possible existence of bad groups; nonsolvable connected groups of finite Morley rank all of whose proper definable subgroups are nilpotent-by-finite. Cherlin had already shown that a simple nonbad group of Morley rank 3 is algebraic \((PSL_ 2(K))\). Poizat shows that bad groups can also give rise to bad fields (a field of finite Morley rank which is not a pure field). After proving Macintyre’s theorem that an \(\omega\)-stable field is algebraically closed, Poizat shows how a definable field comes from a definable action of an abelian group on another abelian group, all within a context of finite Morley rank. Pushing the argument a little more yields Nesin’s theorem: the derived group of a solvable connected group of finite Morley rank is nilpotent. Poizat shows how his method of proof yields the classical Lie-Kolchin theorem. Cherlin’s results on groups of rank 2, 3 are proved by means of studying the definable action of a connected group G on a strongly minimal set X. The possibilities were classified by Hrushovski; the rank of G is \(\leq 3\). If \(rank=2\), G is AGL(2,K) acting on the affine line, if \(rank=3\), G is PGL(2,K) acting on the projective line. Finally strange properties of bad groups are deduced, following Nesin.

Chapter 4 introduces the language of algebraic geometry and the theory of linear algebraic groups, showing how 1950’s algebraic geometry was really 1980’s model theory. Nice proofs are given of the elementary results: an affine algebraic group is linear, a centreless algebraic group is linear. Some space is devoted to proving that a group definable in a (pure) algebraically closed field K is definably isomorphic to an algebraic group over K. This theorem is attributed to Weil-Hrushovski, although should be to van den Dries (at least in characteristic 0). Finally the so-called Borel-Tits theorem is proved. This is rather a misnomer, the Borel-Tits theorem being concerned with isomorphism between rational points of algebraic groups. The new contribution here is rather that the geometric structure of a simple algebraic group over an algebraically closed field is first order definable in the abstract group structure.

Chapter 5 deals with the “pure” theory of stable groups, a theory basically due to Poizat himself. A new and elegant definition of generic types is given. Various results of Hrushovski are proved: an infinitely definable group (living in a stable structure) is enveloped by a definable group; a generically given group is generically isomorphic to an infinitely definable group. (The latter “generalises” a theorem of Andre Weil); a unidimensional stable theory is superstable (no group is mentioned in the hypothesis or conclusion, but binding groups play an essential role in the proof).

In Chapter 6, the Berline-Lascar theory of superstable groups is expounded; a simple superstable group has monomial U rank; the \(\alpha\)- indecomposability theorem, etc.

Chapter 7 is called “Poids” and is devoted largely to work of Hrushovski. The finite rank and superstable theory are simultaneously generalized by the theory of local weight, with respect to some regular type p. A “p-internal, p-connected” group has a finite “p-weight”. Anything one can prove about groups of finite Morley rank goes over to p- internal p-connected groups, replacing rank everywhere by ‘p-weight’ (Hrushovski more generally considers p-simple and p-semiregular groups). Moreover, by another result of Hrushovski a simple group or a field whose generic is nonorthogonal to a (regular) type p is p-internal and p- connected. Thus for example one sees that a stable field whose generic is nonorthogonal to a regular type is algebraically closed.

The illustrations “gentiment porno” which are interspersed throughout the book are somewhat out of place. For this kind of book they should be a little harder.

Reviewer: A.Pillay