On Stable Quotients

Abstract

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then $G^{\mathrm{st}}=G^{00}$ (where $G^{\mathrm{st}}$ is the smallest ∧-definable subgroup with $G∕G^{\mathrm{st}}$ stable, and $G^{00}$ is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion $T^{\mathrm{heq}}$. The second result is an example of a group G definable in a nondistal NIP theory for which $G=G^{00}$, but $G^{\mathrm{st}}$ is not an intersection of definable groups. Our example is a saturated extension of $(\mathbb{R},+,[0,1])$. Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic.