Generic expansions and the group configuration theorem

We exhibit a connection between geometric stability theory and the classification of unstable structures at the level of simplicity and the ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_1$-${\mathrm{S}\mathrm{O}\mathrm{P}}_3$ 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 ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_4$ theory, and use one of the main results of geometric stability theory, the $\mathit{\text{group}}\mathit{\text{configuration}}\mathit{\text{theorem}}$ of Hrushovski, to give an exact correspondence between the geometry of $R$ and the classification-theoretic complexity of $T^R$. Namely, $T^R$ is ${\mathrm{S}\mathrm{O}\mathrm{P}}_3$, and ${\mathrm{T}\mathrm{P}}_2$ 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 ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_1$ (in the equivalence relation version.) This gives us new examples of strictly ${\mathrm{N}\mathrm{S}\mathrm{O}\mathrm{P}}_1$ theories.