# Definably compact groups definable in real closed fields.II

## TL;DR

This paper advances the understanding of definably compact groups in real closed fields by establishing connections with algebraic groups and their universal covers, especially focusing on abelian cases.

## Contribution

It extends previous work by showing that abelian definably compact groups have universal covers isomorphic to certain open subgroups of algebraic group covers.

## Key findings

- For abelian groups, the o-minimal universal cover is isomorphic to a subgroup of an algebraic group's cover.
- The paper links definably compact groups with algebraic groups via definable maps acting as homomorphisms.
- It provides a structural description of the universal cover of abelian definably compact groups.

## Abstract

We continue the analysis of definably compact groups definable in a real closed field $\mathcal{R}$. In [3], we proved that for every definably compact definably connected semialgebraic group $G$ over $\mathcal{R}$ 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. 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 $\widetilde{H\left(R\right)^{0}}$ of the group $H\left(R\right)^{0}$ for some connected $R$-algebraic group $H$.