Externally definable quotients and NIP expansions of the real ordered additive group

Abstract:

Let $\mathscr {R}$ be an $\mathrm {NIP}$ expansion of $(\mathbb {R},<,+)$ by closed subsets of $\mathbb {R}^n$ and continuous functions $f : \mathbb {R}^m \to \mathbb {R}^n$. Then $\mathscr {R}$ is generically locally o-minimal. This follows from a more general theorem on $\mathrm {NIP}$ expansions of locally compact groups, which itself follows from a result on quotients of definable sets in $\aleph _1$-saturated $\mathrm {NIP}$ structures by equivalence relations which are both externally definable and $\bigwedge$-definable. We also show that $\mathscr {R}$ is strongly dependent if and only if $\mathscr {R}$ is either o-minimal or $(\mathbb {R},<,+,\alpha \mathbb {Z})$-minimal for some $\alpha > 0$.