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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 .

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