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 (where is the smallest ∧-definable subgroup with stable, and 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 . The second result is an example of a group G definable in a nondistal NIP theory for which , but is not an intersection of definable groups. Our example is a saturated extension of . 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.
"On Stable Quotients." Notre Dame J. Formal Logic 63 (3) 373 - 394, August 2022. https://doi.org/10.1215/00294527-2022-0023