Even and Mixed Type: Altinel's Program

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

Altinel's program aims at the same result without an inductive hypothesis:

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

The idea is to use model theoretic work of Wagner and the theory of "good tori" to make up for the nonsolvability of potential degenerate sections.

Complete as of Fall 2004; this page reflects the situation in June 2003.

Elimination of mixed Type

Step I
A simple group of mixed type and finite Morley rank must involve a nonalgebraic simple group of even type.

This is found in the paper On groups of finite Morley rank of Even type, Altinel/Cherlin, Journal of Algebra, to appear. Since this reduces mixed type to even type, the rest of the program concerns even type exclusively, and is modelled closely on the previous analysis in the K* case.

The first and stage of that analysis (strong and weak embedding) relied on methods that do not transfer readily to our context, and present the main challenge to the program; the second stage also made use of some properties of K*-groups in what seemed at the time an essential way.

All of these issues can however be overcome.

Step II
Classification of groups of even type with weakly embedded subgroups.
As this blows up relative to earlier work, we divide the analysis into four parts:

• Weak solvability
On groups of finite Morley rank of Even type, Altinel/Cherlin, Journal of Algebra 264 (2003), 155-185.
© Acad. Press.
Also contains the results in mixed type given above.
• Strong embedding, case I
Simple L*-groups of finite Morley rank of even type with strongly embedded subgroups, Altinel/Cherlin, Journal of Algebra 272 (2004), 95-127.
© Acad. Press.
• Weak embedding, case I
Simple L*-groups of finite Morley rank of even type with weakly embedded subgroups, Altinel/Cherlin, J. Algebra, to appear.
© Acad. Press.
• Strong and Weak embedding, case II
Limoncello, Altinel/Cherlin, J. Algebra, to appear.
© Acad. Press.

Step III and beyond
Redoing the strongly closed abelian case, then arguing that the remainder of the argument (which is still quite long) can be carried out as in the even type case, with certain adjustements.

A projected book by Altinel, Borovik, Cherlin will carry out the analysis in full. After the strongly closed abelian case it largely rejoins the previous analysis.

Note
An L*-group is, for our purposes, a group of even type, whose proper definable simple sections of nondegenerate type are algebraic. By working with these groups rather than with K*-groups, we arrange that the classification of groups of even type can be carried out in an inductive framework.

• Groups of finite Morley rank
• Odd and degenerate types.
• Even and Mixed types.