Definable structures in o-minimal theories: One dimensional types

Abstract

Let N be a structure definable in an o-minimal structure M and p ∈ S N (N), a complete N-1-type. If dim M (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.