2017/03/24 by Barriga, Eliana
#03C64 #03C68 #20G20 #22B99 #22E15 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1703.08606
We study definably compact definably connected groups definable in a sufficiently saturated real closed field R. We introduce the notion of group-generic point for \bigvee-definable groups and show the existence of group-generic points for definably compact groups definable in a sufficiently saturated o-minimal expansion of a real closed field. We use this notion along with some properties of generic sets to prove that for every definably compact definably connected group G definable in 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. This result is used in [2] to prove that the o-minimal universal covering group of an abelian connected definably compact group definable in a sufficiently saturated real closed field R is, up to locally definable isomorphisms, an open connected locally definable subgroup of the o-minimal universal covering group of the R-points of some R-algebraic group.