Groups definable in weakly o-minimal non-valuational structures

Abstract

Let $\mathcal{M}$ be a weakly o-minimal non-valuational structure, and $\mathcal{N}$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $\mathcal{M}$ is a subgroup of a group $K$ definable in $\mathcal{N}$, which is canonical in the sense that it is the smallest such group. As an application, we obtain that $G^{00}=G\cap K^{00}$, and establish Pillay's Conjecture in this setting: $G/G^{00}$, equipped with the logic topology, is a compact Lie group, and if $G$ has finitely satisfiable generics, then ${\dim }_{Lie}(G/G^{00})=\dim (G)$.