The Even and Mixed Type Theorems

Theorem
Let G be a simple K*-group of finite Morley rank which is of mixed or even type.
Then G is algebraic.

Remarks
The hypothesis can be phrased as follows: G contains a nontrivial 2-unipotent subgroup. Note that according to the conclusion, G cannot be of mixed type.

This page contains some documentation relating to the proof of the theorem above.

Elimination of mixed Type

In two papers it is shown that K* groups of finite Morley rank are not of mixed type. The first paper treats the tame case, and the second handles the general case.

T. Altinel, A. Borovik, G. Cherlin

Groups of mixed type, J. Alg 192 (1997), 524-571.

E. Jaligot

Groupes de type mixte, J. Alg 212 (1999), 753-768.

Classification of Even Type Groups

Final paper

The "final" paper, giving the classification of the simple K*-groups of finite Morley rank of even type, simply sums up the results of a long series of papers on the topic. It can be taken as a guide to the relevant literature.

T. Altinel, A. Borovik, and G. Cherlin

K*-groups of finite Morley rank and of even type, preprint

In the proof, one distinguishes three cases - thin, quasithin, and generic - corresponding ultimately to groups of Lie rank 1, 2, or greater than 2, respectively.

See below for further developments arising from Altinel's habilitation.

The main ingredients of the analysis are found in the papers listed below.

Preparation

Strong and weak embedding

This involves two versions of Bender's classification, involving strongly and weakly embedded subgroups. The treatment in the tame case is in

T. Altinel

Groups of finite Morley rank with strongly embedded subgroups, J. Algebra 180 (1996), 778-807

T. Altinel, A. Borovik, G. Cherlin

On groups of finite Morley rank with weakly embedded subgroups, J. Algebra 211 (1999), 409-456

The general case is treated in

E. Jaligot

Groupes de type pair avec un sous-groupe faiblement inclus, J. Algebra 240 (2001), 413-444.

Parabolic structure

The weak embedding theorem can be used to eliminate cores of 2-local subgroups, in particular in parabolic subgroups. The next step is to eliminate components of parabolic subgroups. First one classifies groups with strongly closed abelian subgroups (and, in the process, one eliminates components of type SL₂). Then pushing-up and a global C(G,T) theorem are proved, and finally the components are eliminated.

T. Altinel, A. Borovik, and G. Cherlin

Groups of finite Morley rank with strongly closed abelian subgroups, J. Algebra, J. Alg. 232 (2000), 420-461.

T. Altinel, A. Borovik, and G. Cherlin

Pushing up and C(G,T) in groups of finite Morley rank of even type, J. Alg. 247 (2000), 541-576.

T. Altinel, A. Borovik, G. Cherlin, and L.-J. Corredor

Parabolic 2-local subgroups in groups of finite Morley rank of even type, J. Alg., to appear.

J. Algebra 269 (2003), 250-2662

Recognition

Depending on the number of minimal parabolic subgroups, one either applies the global C(G,T) Theorem directly, or one arrives at either a rank 2 amalgam or a situation covered by Niles' theorem and generic identification theorems. This process is described in detail in the " final" paper.

The thin case

In this case there are no proper minimal parabolic subgroups, and the global C(G,T) theorem applies to show that the group is SL₂.

The quasithin case

This uses the amalgam method of Delgado and Stellmacher. The argument is lengthy, and much of it follows a paper by Stellmacher quite closely. We intend to give full details either in a Dimacs technical report, or a book.
The argument is outlined in the following article.

T. Altinel, A. Borovik, G. Cherlin

Classification of simple K*-groups of finite Morley rank and even type: Geometric aspects,
in Groups, Combinatorics, and Geometry (Durham 2001)

, pp. 1-12, World Sci. Publ., River Edge, NJ 2003.

As is explained in that account, the amalgam method by itself does not identify the group, but constructs a Moufang generalized n-gon for some n, having the same data. At this point, one invokes the classification of Moufang generalized n-gons of finite Morley rank to identify the "data". This classification is given in

L. Kramer, K. Tent, H. van Maldeghem

Simple groups of finite Morley rank and Tits buildings,
Israel J. Mathematics 109 (1999), 189-224.

Then one applies a recognition theorem of Tits, for which Bennett and Shpecterov have recently given an elegant proof.

The generic case

This is handled by an analog of a theorem of Niles:

A. Berkman and A. Borovik

An identification theorem for groups of finite Morley rank and even type

J. Algebra 266 (2003), 375-381.

This also uses the results of Kramer, Tent, and van Maldeghem cited above, in Tits rank at least 3.
Another approach is developed in

A. Berkman and A. Borovik

A generic identification theorem for groups of finite Morley rank, submitted.

This aims directly at Lyons' generalization of the Curtis-Phan theorem.

Altinel's Program

In Altinel's habilitation at Lyon he proposed the elimination of the K*-hypothesis throughout the above. This is a sweeping proposal which is under exploration. It is known at this time that Wagner's results on fields of finite Morley rank provide an essential tool which can replace assumptions of solvability at some critical points.

More here