Geometric structures with a dense independent subset

Abstract

We study a generalization of the expansion by an independent dense set, introduced by Dolich, Miller, and Steinhorn in the o-minimal context, 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 $T^\mathrm{ind}$, 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 SU-rank 1, or superrosy of thorn-rank 1, then $T^\mathrm{ind}$ is $\omega$-stable, supersimple, and superrosy, respectively, and its U-/SU-/thorn-rank is either 1 (if T is trivial) or $\omega$ (if T is non-trivial). In the supersimple SU-rank 1 case, we obtain a description of forking and canonical bases in $T^\mathrm{ind}$. We also show that if T is (strongly) dependent, then so is $T^\mathrm{ind}$, and if T is non-trivial of finite dp-rank, then $T^\mathrm{ind}$ has dp-rank greater than n for every $n<\omega$, but bounded by $\omega$. In the stable case, we also partially solve the question of whether any group definable in $T^\mathrm{ind}$ comes from a group definable in T.