Pantelis Eleftheriou (University of Leeds)

Pillay’s Conjecture for groups definable in weakly o-minimal non-valuational structures

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Let $M$ be a weakly o-minimal non-valuational structure, and $N$ its canonical o-minimal extension (by Wencel). We prove that every group $G$ definable in $M$ is a dense subgroup of a group $K$ definable in $N$. 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 $\dim(G/G^{00})= \dim(G)$.