Type-definable and invariant groups in o-minimal structures

Abstract

Let $M$ be a big o-minimal structure and $G$ a type-definable group in $M^n$. We show that $G$ is a type-definable subset of a definable manifold in $M^n$ that induces on $G$ a group topology. If $M$ is an o-minimal expansion of a real closed field, then $G$ with this group topology is even definably isomorphic to a type-definable group in some $M^k$ with the topology induced by $M^k$. Part of this result holds for the wider class of so-called invariant groups: each invariant group $G$ in $M^n$ has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as $M^n$.