We consider groups definable in the structure ℝan 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 Rn.
ASJC Scopus subject areas
- Mathematics (all)