# About the mini-course

**Groups of finite Morley rank**are the model-theorist’s approach to algebraic groups over algebraically closed fields: one has a notion of dimension (the Morley rank) which behaves like the Zariski dimension, but there is no given topological/functorial structure. This allows only basic arguments. It was conjectured by Cherlin and Zilber that all infinite simple groups of finite Morley rank ought to be algebraic groups over algebraically closed fields. And despite nearly 40 years of efforts, the Cherlin-Zilber conjecture is still open.

As a matter of fact, the solution in rank 3 was given only two years ago by Frécon. It is completely independent from the bulk of earlier work (mostly cfsg-oriented), and it is unclear whether it will open new paths. But Wagner’s rewriting of Frécon’s theorem is a beautiful and significant piece of mathematics, which can be taught in a self-contained class.

**This mini-course**will thus explain the full solution of the conjecture in rank 3. It is of interest to at least four (non-disjoint) kinds of mathematicians:

**Model-theorists:**who will understand what is going on since no big group-theoretic guns are required in small rank.**Group-theorists:**who will enjoy seeing how important involutions are even in mathematical logic.**Geometers:**who will be puzzled by what happens when one looks at algebraic groups from the model-theoretic perspective.**Aesthetes:**since, needless to say, the methods are beautiful.

**Tentative syllabus for 5 lectures of 120 minutes each:**

- Ranked universes: algebraic structures bearing a decent dimension (à la Borovik-Poizat). Groups of finite Morley rank as groups living in ranked universes.
- The Cherlin-Zilber “algebraicity” conjecture: an infinite simple ranked group ought to be of the form G(K) for G a simple algebraic group and K an algebraically closed field. This will be motivated by the natural examples, and by Macintyre’s theorem on fields of finite Morley rank.
- State of the art on the Cherlin-Zilber: what works and what doesn’t.

Involutions (elements of order 2) as the main dividing line. The lack of a Feit-Thompson theorem. The Altınel-Borovik-Cherlin “at least even type” theorem. And our limited knowledge in “at most odd type”. (A catalogue of nightmares, only if time permits.)

- The Descending Chain Condition on definable subgroups (an Artin-like condition) — and as a consequence, the notion of an ill-named “connected component”.
- Action of a connected group on a finite set.
- Reineke’s theorem: a connected group of rank 1 is abelian.

Remarkably the proof already involves involutions. - One can still classify isomorphism types of such groups (Macintyre’s theorem on abelian groups, proved only if time permits).

- Cherlin’s Theorem: a connected group of rank 2 is soluble.

Here again, involutions play a crucial role and the theory decidedly meets finite group theory. - Although the non-nilpotent case can be determined (using linearisation), one cannot classify the nilpotent ones. Time permitting I may mention the Baudisch groups, and discuss categoricity results.

- Nesin’s Theorem: a simple group of rank 3 non-isomorphic to PGL2(K) has no involutions.

The proof is pure projective geometry. It was generalised by Corredor to higher rank. - Most importantly I shall make a point not to invoke “Bachmann’s Theorem”.
- Actually the theorem has been generalised by Wiscons and myself (work in progress) to a nice class of groups. Time permitting I may describe this and further conjectures.

- Frécon’s theorem: a simple group of rank 3 is isomorphic to PGL2(K).

We shall give our compactification with Corredor of Wagner’s version of Frécon’s Theorem. It essentially builds on the tightness of the configuration to construct an involutive automorphism, something known impossible for years. - I may discuss what is known in ranks 4 and 5 (which are at present open).

# Lectures

**Classroom:**

- October 16, 22, 23, 25: Hemiciclo LL-001.

- October 18: ML-617.

**Organizer: **Luis Jaime Corredor - lcorredo@uniandes.edu.co