Algebraicity conjecture

Cherlin/Zilber

Conjecture   A simple group of finite Morley rank is algebraic.

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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

Survey Talks (Slides only)


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

Home Pages
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Tuna Altinel
Eric Jaligot

This material is based in part upon work supported by the NSF under Grant No. 0100794 and earlier grants. Any opinions, findings, conclusions, or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundaion.

This page © T. Altinel, A. Borovik, and G. Cherlin.
Errors and omissions © G. Cherlin.