Title: Definably compact groups definable in real closed fields. I,
Abstract: 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 $\phi$ from a generic
definable neighborhood of the identity of $G$ into the group $H\left(R\right)$
of $R$-points of $H$ such that $\phi$ 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.