## Abstract

We consider groups definable in the structure ℝ_{an} and certain o-minimal expansions of it. We prove: If dubble script G sign = 〈G, *〉 is a definable abelian torsion-free group, then dubble script G sign is definably isomorphic to a direct sum of (ℝ, +)^{k} and 〈ℝ^{>0},·〈^{m}, for some k,m ≥ 0. Futhermore, this isomorphism is definable in the structure 〈ℝ, +, -, dubble script G sign〉. In particular, if dubble script G sign is semialgebraic, then the isomorphism is semialgebraic. We show how to use the above result to give an "o-minimal proof to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero. We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and dubble script G sign a definable group of dimension n. The group dubble script G sign is torsion-free if and only if dubble script G sign, as a definable group-manifold, is definably diffeomorphic to R^{n}.

Original language | English |
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Pages (from-to) | 1299-1321 |

Number of pages | 23 |

Journal | Illinois Journal of Mathematics |

Volume | 49 |

Issue number | 4 |

DOIs | |

State | Published - 2005 |

## ASJC Scopus subject areas

- Mathematics (all)