No abstract

We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We characterize linear" theories in terms of properties of the corresponding theory of the lovely pair. For o-minimal theories, we use Peterzil-Starchenko's trichotomy theorem to characterize for suciently general points, the local geometry around it in terms of the thorn U rank of its type.

Abstract. We generalize the work of [13] on expansions of o-minimal structures with dense independent subsets, to the setting of geometric structures. We introduce the notion of an H-structure of a geometric theory T , show that H-structures exist and are elementarily equivalent, and establish some basic properties of the resulting complete theory T ind , including quantifier elimination down to "H-bounded" formulas, and a description of definable sets and algebraic closure. We show that if T is strongly minimal, supersimple of SUrank 1, or superrosy of thorn rank 1, then T ind is ω-stable, supersimple, and superrosy, respectively, and its U-/SU-/thorn rank is either 1 (if T is trivial) or ω (if T is non-trivial). In the supersimple SU-rank 1 case, we obtain a description of forking and canonical bases in T ind . We also show that if T is (strongly) dependent, then so is T ind , and if T is non-trivial of finite dp-rank, then T ind has dp-rank greater than n for every n < ω, but bounded by ω. In the stable case, we also partially solve the question of whether any group definable in T ind comes from a group definable in T .

Abstract. We define and study the notion of ample metric generics for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal in [KR07]. Our work is based on the concept of a Polish topometric group, defined in this article. Using Kechris and Rosendal's work as a guide, we explore consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group Iso(U 1 ) of the bounded Urysohn space, the unitary group U (ℓ 2 ) of a separable Hilbert space, and the automorphism group Aut([0, 1], λ) of the Lebesgue measure algebra on [0, 1]. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from Aut([0, 1], λ) into a separable topological group is continuous.

We characterize þ-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.