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Groupes géométriques de rang de Morley fini

Part of: Model theory Foundations

Published online by Cambridge University Press:  07 October 2008

Olivier Frécon
Olivier Frécon
Affiliation:
Laboratoire de Mathématiques et Applications, Université de Poitiers, Téléport 2 – BP 30179, Boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil Cedex, France ([email protected])
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Abstract

We consider a new subgroup In(G) in any group G of finite Morley rank. This definably characteristic subgroup is the smallest normal subgroup of G from which we can hope to build a geometry over the quotient group G/ In(G). We say that G is a geometric group if In(G) is trivial.

This paper is a discussion of a conjecture which states that every geometric group G of finite Morley rank is definably linear over a ring K1 ⊕…⊕ Kn where K1,…,Kn are some interpretable fields. This linearity conjecture seems to generalize the Cherlin–Zil'ber conjecture in a very large class of groups of finite Morley rank.

We show that, if this linearity conjecture is true, then there is a Rosenlicht theorem for groups of finite Morley rank, in the sense that the quotient group of any connected group of finite Morley rank by its hypercentre is definably linear.

Keywords

groups of finite Morley rankalgebraic groupsuniformly definable families of subgroupsCarter subgroups

MSC classification

Secondary: 03C45: Classification theory, stability and related concepts 03C60: Model-theoretic algebra 20A15: Applications of logic to group theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 7 , Issue 4 , October 2008 , pp. 751 - 792
Copyright
Copyright © Cambridge University Press 2008

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