Definable groups in dense pairs of geometric structures

Abstract

We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large (in the sense of dimension over the predicate) definable subgroups (the new language) of groups definable in the original language $\mathcal {L}$ are also $\mathcal {L}$-definable, and definably amenable $\mathcal {L}$-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group definable in a model of T, then the (type-definable) connected component $G_{\mathcal {L}_P}^{00}$ of G in the expansion agrees with the connected component $G^{00}$ in the original language. We prove similar preservation results for $G\cap P(M)^n$, the P-part of G in a lovely pair (M, P), and for subgroups of G in pairs (R, G) where R is a real closed field and G is a subgroup of $R^{>0}$ or the unit circle ${\mathbb {S}}(R)$ with the Mann property.