On stable quotients

Abstract

We solve two problems from the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay, which concern maximal stable quotients of groups type-definable in NIP theories. The first result says that if $G$ is a type-definable group in a distal theory, then $G^{st}=G^{00}$ (where $G^{st}$ is the smallest type-definable subgroup with $G/G^{st}$ stable, and $G^{00}$ is the smallest type-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from $T$ to the hyperimaginary expansion $T^{heq}$. The second result is an example of a group $G$ definable in a non-distal, NIP theory for which $G=G^{00}$ but $G^{st}$ 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.