# Locally definable subgroups of semialgebraic groups

## TL;DR

This paper proves a conjecture about the structure of semialgebraic groups over real closed fields, showing that generated groups from definable sets contain generic sets and are divisible if connected.

## Contribution

It establishes that in abelian semialgebraic groups, the group generated by a definable set contains a generic set and is divisible if connected, extending to o-minimal expansions.

## Key findings

- Generated groups contain a generic set.
- Connected generated groups are divisible.
- The result applies to o-minimal expansions of real closed fields.

## Abstract

We prove the following instance of a conjecture stated in arXiv:1103.4770. Let $G$ be an abelian semialgebraic group over a real closed field $R$ and let $X$ be a semialgebraic subset of $G$. Then the group generated by $X$ contains a generic set and, if connected, it is divisible. More generally, the same result holds when $X$ is definable in any o-minimal expansion of $R$ which is elementarily equivalent to $\mathbb R_{an,exp}$. We observe that the above statement is equivalent to saying: there exists an $m$ such that $\Sigma_{i=1}^m(X-X)$ is an approximate subgroup of $G$.