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uniformly definable families of subgroups

Journal of the Institute of Mathematics of Jussieu (1)

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1 results

Groupes géométriques de rang de Morley fini
Part of
- Model theory
- Foundations
Olivier Frécon
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 7 / Issue 4 / October 2008

Published online by Cambridge University Press:

07 October 2008, pp. 751-792

Print publication:

October 2008
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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.