On Valued C-minimal Groups

Patrick Simonetta

D. Macpherson and C. Steinhorn introduced some variants of the notion of o-minimality. One of the most interesting ones is C-minimality, which provides a natural setting for studying algebraically closed valued fields and some valued groups. We have gone further in the study of the structure of C-minimal valued groups, giving a partial characterization in the abelian case, and some new structure theorems about the non-abelian case. In the abelian case we obtain the following principle: let G be an abelian valued group for which the valuation satisfies some kind of compatibility with the multiplication by any prime number p. Then G is C-minimal if and only if the expansion of the chain of valuations I by the maps induced by the multiplication by each prime number in G, and by some unary predicates controlling the cardinality of the residual structures, is o-minimal. Using this principle we can construct many new examples of abelian C-minimal valued groups. Our main progress in the non-abelian case concerns the case where all the residual structures are infinite: we prove that the commutator function of the group induces a function from I × I to I whose partial maps are increasing, and satisfy the intermediate value property, and some ”bilinear” functions between residual structures. In [1] it was shown that any C-minimal valued group 𝔾 contains a definable abelian normal subgroup A such that G/A is of finite exponent. We can now improve this result under our assumption on residual structures: we prove then that G/A is a nilpotent group of exponent a power of a prime number p.

[1] H. D. Macpherson, C. Steinhorn,On variants of o-minmality, Annals of Pure and Applied Logic 79 (1996), 165-209.

[2] P. Simonetta, Abelian C-minimal groups, To appear in Annals of Pure and Applied Logic.

[3] P. Simonetta, On non-abelian C-minimal groups, in preparation.

Address: Equipe de Logique Mathématique de Paris VII 2, place Jussieu - case 7012 - 75251 PARIS cedex 05 - FRANCE

E-mail: simbaud@logique.jussieu.fr