Definable Structures in O-minimal Theories: One Dimensional Types

Let N be a structure definable in an o-minimal structureM and p ∈ SN (N), a complete N -1-type. If dimM(p) = 1 then p supports a combinatorial pre-geometry. We prove a Zilber type trichotomy: Either p is trivial, or it is linear, in which case p is non-orthogonal to a generic type in an N -definable (possibly ordered) group whose structure is linear, or, if p is rich then p is non-orthogonal to a generic type of an N -definable real closed field. As a result we obtain a similar trichotomy for definable one-dimensional structures in o-minimal theories. In this paper we prove a trichotomy theorem for one-dimensional types in structures definable in o-minimal theories. With this we conclude the work started in [4], of which this is a direct continuation. Recall that a structure N is said to be definable in an o-minimal structure M if the universe N , of N , as well as all its atomic relations, are definable sets (possibly of several variables, possibly using parameters) in the structure M. In [4] we proved a weak version of Zilber’s trichotomy: Theorem 1. Let N be a stable structure definable in an o-minimal structure M. If dimM(N) = 1 then N is 1-based. The local nature of phenomena in o-minimal theories does not leave room for more precise global statements in the unstable case. The aim of this paper is to remedy this situation by applying the results of [4] and [2] to obtain a complete classification of 1-M-dimensional types in N without any additional global assumptions on N . Our main result can be summed up by (see definitions below): Theorem 2. Let M be an o-minimal structure and N definable in M. Let p ∈ SN (N) be one-M-dimensional. Then exactly one of the following holds: (1) p is trivial. (2) p is linear, in which case p is non-orthogonal to a generic type of an N definable (possibly locally ordered) group G. The structure which N induces on G is linear, i.e., given by definable (possibly local) subgroups of Gn. (3) p is rich, in which case it is non-orthogonal to a generic type of an N definable real closed field R. In fact, our results will be more precise and give a stable/unstable dichotomy (see Theorem 2.1). As a corollary to the above we can complete the analysis of definable one dimensional structures which began in [4]: ∗Supported by the EPSRC grant no. EP C52800X 1. ∗∗ Partially supported by the EC FP6 through the Marie Curie Research Training Network MODNET (MRTN-CT-2004-512234).