Generic expansions and the group configuration theorem

TL;DR

This paper connects geometric stability theory with classification of unstable structures, introducing generic expansions of theories and using the group configuration theorem to classify their complexity.

Contribution

It introduces generic expansions of theories based on definable relations and uses geometric stability theory to classify their model-theoretic complexity.

Findings

$T^{R}$ is $ ext{NSOP}_4$ when $T$ is weakly minimal and $R$ is a ternary fiber algebraic relation.

$T^{R}$ is $ ext{SOP}_3$ or $ ext{TP}_2$ depending on the geometric properties of $R$.

Provides new examples of strictly $ ext{NSOP}_1$ theories.

Abstract

We exhibit a connection between geometric stability theory and the classification of unstable structures at the level of simplicity and the $\mathrm{NSOP}_{1}$-$\mathrm{SOP}_{3}$ gap. Particularly, we introduce generic expansions $T^{R}$ of a theory $T$ associated with a definable relation $R$ of $T$, which can consist of adding a new unary predicate or a new equivalence relation. When $T$ is weakly minimal and $R$ is a ternary fiber algebraic relation, we show that $T^{R}$ is a well-defined $\mathrm{NSOP}_{4}$ theory, and use one of the main results of geometric stability theory, the \textit{group configuration theorem} of Hrushovski, to give an exact correspondence between the geometry of $R$ and the classification-theoretic complexity of $T^{R}$. Namely, $T^{R}$ is $\mathrm{SOP}_{3}$, and $\mathrm{TP}_{2}$ exactly when $R$ is geometrically equivalent to the graph of a type-definable group operation; otherwise, $T^{R}$ is either simple (in the predicate version of $T^{R}$) or $\mathrm{NSOP}_{1}$ (in the equivalence relation version.) This gives us new examples of strictly $\mathrm{NSOP}_{1}$ theories.