Definably compact groups definable in real closed fields. I

TL;DR

This paper investigates definably compact, connected groups in real closed fields, introducing group-generic points and establishing a connection to algebraic groups, advancing the understanding of their structure in o-minimal settings.

Contribution

It introduces the notion of group-generic points for $igvee$-definable groups and proves the existence of a definable homomorphism to an algebraic group for such definably compact groups.

Findings

Existence of group-generic points for definably compact groups.

Construction of a definable injective map to an algebraic group.

Application to universal covering groups in o-minimal structures.

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.