Strongly minimal groups in o-minimal structures

TL;DR

This paper proves Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups definable in o-minimal structures, showing they interpret algebraically closed fields and are isomorphic to algebraic groups.

Contribution

It establishes the conjecture for a new class of structures, linking strongly minimal groups in o-minimal structures to algebraic groups over algebraically closed fields.

Findings

Strongly minimal non-locally modular groups interpret algebraically closed fields.

Such groups are definably isomorphic to algebraic groups over these fields.

The result confirms Zilber's Trichotomy in this setting.

Abstract

We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.