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 L are also L-definable, and definably amenable 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 00 L P of G in the expansion agrees with the connected component G 00 in the original language. We prove similar preservation results for G ∩ 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 S(R) with the Mann property. Keywords Unary predicate expansions • Definable groups • Amenable groups • Mann property • Connected component Mathematics Subject Classification 03C45 • 03C64A. Berenstein was partially supported by Colciencias' Project Teoría de modelos y dinámicas topológicas number 120471250707. E. Vassiliev was partially supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant AP05134992 Conservative extensions, countable ordered models, and closure operators). Both authors were also supported by the project Expansiones densas de teorías geométricas: grupos y medidas, Facultad de Ciencias, Universidad de los Andes. The authors thank Erik Walsberg for providing the proof for Proposition 4.11.

Vector spaces with a dense-codense generic submodule

“…Recall from [5] that a unary expansion (M, P ) of a model M of geometric theory T satisfies Type Equality Assumption (TEA) if whenever a, b, c ∈ M are such that a is P -independent (i. The following corollary follows directly form Corollary 3.21.…”