Tame Expansions of ω-Stable Theories and Definable Groups

Abstract

We study groups definable in tame expansions of $\omega$-stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair $(K,G)$, where $K$ is an algebraically closed field and $G$ is a multiplicative subgroup of $K^{\times }$ with the Mann property, we show that the pair $(K,G)$ satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in $(K,G)$ in terms of algebraic groups in $K$ and interpretable groups in $G$. Furthermore, we compute the Morley rank and the $U$-rank in $(K,G)$ and both ranks agree.