Strongly minimal groups in o-minimal structures

Abstract

We prove Zilber’s Trichotomy Conjecture for strongly minimal expansions of 2-dimensional groups, definable in o-minimal structures:

Theorem. Let $\mathcal{M}$ be an o-minimal expansion of a real closed field, $⟨G;+⟩$ a 2-dimensional group definable in $\mathcal{M}$, and $\mathcal{D}=⟨G;+,\dots ⟩$ a strongly minimal structure, all of whose atomic relations are definable in $\mathcal{M}$. If $\mathcal{D}$ is not locally modular, then an algebraically closed field $K$ is interpretable in $\mathcal{D}$, and the group $G$, with all its induced $\mathcal{D}$-structure, is definably isomorphic in $\mathcal{D}$ to an algebraic $K$-group with all its induced $K$-structure.