Locally definable subgroups of semialgebraic groups

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 $G$.

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