Algebraicity conjecture

Cherlin/Zilber

Conjecture   A simple group of finite Morley rank is algebraic.

On this page we explain the Borovik Program

The four types

K*-groups

Mixed and even type

Odd and degenerate type

The Four Types

The connected component of a Sylow 2-subgroup of a group of finite Morley rank is a central product U*T with U 2-unipotent and T a 2-torus; the intersection of U and T is finite.

In algebraic groups the 2-elements are unipotent if the characteristic is 2, and semisimple otherwise; so in characteristic 2 one expects T=1, and otherwise one expects U=1. However, a product of finitely many algebraic groups of varying characteristics also has finite Morley rank.

Accordingly, we consider four types of groups of finite Morley rank, corresponding to the possible structures of the Sylow 2-subgroup:

	T>1	T=1
U>1	Mixed Type	Even type
U=1	Odd Type	Degenerate Type

Or more explicitly:

Even	Odd	Mixed	Degenerate
So=U Unipotent	So=T Torus	So=U*T	So=1 S Finite

K-groups and K*-groups

A K-group is one whose definable infinite simple sections are algebraic.

A K*-group is one whose proper definable infinite simple sections are algebraic.

A minimal counterexample to the Algebraicity Conjecture would be a K*-group; conversely, a K*-group is either a K-group or a minimal counterexample to the algebraicity conjecture.

Results

The structure of the Sylow 2-subgroup in a simple K*-group of finite Morley rank has been substantially clarified. The critical distinction is between the cases U>1 (mixed and even type) and U=1 (odd and degenerate types).

Mixed and Even Type

In the mixed and even type cases, any simple K*-group of finite Morley rank is algebraic.

More generally, there are no simple groups of finite Morley rank of mixed type, and any simple group of finite Morley rank is algebraic.

(Details, mixed and even type.)

Odd Type

In the odd type case, any simple K*-group of finite Morley rank is either algebraic, or is minimal connected simple with Prüfer 2-rank at most two.

Degenerate Type

In the degenerate K* case, the normal 2-rank is at most 2 or there is a proper 2-generated core.

(Details, odd and degenerate type.)

Surveys

Overview

---

• A Brief Introduction to groups of Finite Morley Rank
A. Borovik; based on a talk given in 1994
21 pp. Introductory. Tame conjecture in Section 4
• Tame groups of Odd and Even Type
A. Borovik, November 1997; appeared 1998 in Carter/Saxl Algebraic Groups and their Representations, Kluwer
27 pp. Full statement of the program. Even type case was expected to use component analysis. Most of this was bypassed.
• Simple groups of finite Morley rank, 2002
G. Cherlin, Spring 2002, Ravello. To appear 2005 in proceedings volume.
17 pp. Emphasizes Burdges' work eliminating tameness hypotheses in odd type (somewhat oversimplified, cf. Burdges' dissertation, Rutgers 2004, Chap. 10) and Altinel's approach to even and mixed type, then at an early stage.

---

Even Type

---

• Simple groups of finite Morley rank of even type
Tuna Altinel, Fall 2000 (this copy shows a printing date of 2002)
22 pp. The classification of even type K*-groups.
• Classification of the simple groups of finite Morley rank
Tuna Altinel, Contemporary Mathematics 302, 2002, 121-147
26 pages. After discussing even type, also enters into the theory of solvable groups, degenerate type (and relations to Feit-Thompson) as well as his (new) approach to even type without K* hypotheses.

Survey Talks (Slides only)

Overview

---

• Simple groups of finite Morley rank
  G. Cherlin, ASL Annual Meeting 2003
  25 pages, slides only.

  ---

  Odd Type

  ---

• Tutorial at Groups, Geometry, and Logic, CIRM, Luminy 2004
September 27
• Simple groups of finite Morley rank, 2004
G. Cherlin
22 slides (68 overlays). Overview
September 28
• A generic identification theorem for the classification projet of simple groups of finite Morley rank
Ayse Berkman
14 slides. Generic identification for odd type.
September 29
• 0-Unipotence in odd type groups of finite Morley rank
Jeff Burdges
34 slides. Elimination of tameness hypotheses from the analysis of generic odd type groups.
• Sousgroupes de Carter et groupes résolubles de rang de Morley fini">
Olivier Frécon
31 slides. Solvable groups
September 30
• Minimal simple groups of finite Morley rank
Eric Jaligot
24 slides. Minimal connected simple groups, bound on Prüfer rank
October 1
• Groups of finite Morley rank - Lessons learned
Alexandre Borovik
45 pp. Problems in finite group theory suggested by this work.

Reference: Borovik/Nesin, Groups of Finite Morley Rank, Oxford Logic Guides 26.

Home Pages
	Alexandre Borovik Tuna Altinel Eric Jaligot