By Justin Moore
By Filippo Calderoni
By Sandra Müller
Appears in collection : Model theory of valued fields / Théorie des modèles des corps valués
In this talk there is no valued field but we try to find one. Or, to be more modest, we try first to find a group. Our problematic is the trichotomy of Zilber. Given an abstract structure which shares certain model theoretical properties with an infinite group (or with an infinite field) can we define an infinite group (or an infinite field) in this structure? The initial conjecture was about strongly minimal structures and it turned out to be wrong. It becomes correct in the framework of Zariski structures. These are minimal structures in which some definable sets are identified as closed, the connection between closed and definable sets being similar to what happens in algebraically closed fields with the topologies of Zariski. This is the content of a large volume of work by Ehud Hrushovski and Boris Zilber. O-minimal structures and their Cartesian powers arrive equipped with a topology. Although these topologies are definitely not Noetherian, the situation presents great analogies with Zariski structures. Now, Kobi Peterzil and Sergei Starchenko have shown Zilber's Conjecture in this setting (up to a nuance). The question then arises naturally in $C$-minimal structures. Let us recall what they are. $C$-sets can be understood as reducts of ultrametric spaces: if the distance is $d$, we keep only the information given by the ternary relation $C(x, y, z)$ iff $d(x, y)=d(x, z)>d(y, z)$. So, there is no longer a space of distances, we can only compare distances to a same point. A $C$-minimal structure $M$ is a $C$-set possibly with additional structure in which every definable subset is a Boolean combination of open or closed balls, more exactly of their generalizations in the framework of $C$-relations, cones and 0-level sets. Moreover, this must remain true in any structure $N$ elementary equivalent to $M$. Zilber's conjecture only makes sense if the structure is assumed to be geometric. Which does not follow from $C$-minimality. Nearly 15 years ago Fares Maalouf has shown that an inifinite group is definable in any nontrivial locally modular geometric $C$-minimal structure. Fares, Patrick Simonetta and myself do the same today in a non-modular case. Our proof draws heavily on that of Peterzil and Starchenko.