## Abstract

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

Theorem. Let M be an o-minimal expansion of a real closed field, hGI Ci a 2-dimensional group definable in M, and D D hGI C; : : :i 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

Theorem. Let M be an o-minimal expansion of a real closed field, hGI Ci a 2-dimensional group definable in M, and D D hGI C; : : :i 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

Original language | English GB |
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Pages (from-to) | 3351-3418 |

Number of pages | 68 |

Journal | Journal of the European Mathematical Society |

Volume | 23 |

Issue number | 10 |

State | Published - 2021 |