Let N be a structure definable in an o-minimal structure M 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.
Bibliographical noteFunding Information:
∗ Supported by the EPSRC grant no. EP C52800X 1 ∗∗Partially supported by the EC FP6 through the Marie Curie Network MODNET (MRTN-CT-2004-512234) Received April 14, 2007 and in revised form January 1, 2009
ASJC Scopus subject areas
- Mathematics (all)