Submission date: 21 May 2017

Abstract:

We continue the analysis of definably compact groups definable in a real closed field R. In [3], we proved that for every definably compact definably connected semialgebraic group G over R there are a connected R-algebraic group H, a definable injective map φ from a generic definable neighborhood of the identity of G into the group H(R) of R-points of H such that φ acts as a group homomorphism inside its domain. The above result and our study of locally definable covering homomorphisms for locally definable groups combine to prove that if such group G is in addition abelian, then its o-minimal universal covering group \widetilde{G} is definably isomorphic, as a locally definable group, to a connected open locally definable subgroup of the o-minimal universal covering group \widetilde{H(R)^{0}} of the group H(R)^{0} for some connected R-algebraic group H.

Mathematics Subject Classification: 03C64, 20G20, 22E15, 03C68, 22B99

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Full text arXiv 1705.07370: pdf, ps.