Submission date: 6 October 2021

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 Gst=G00 (where Gst is the smallest type-definable subgroup with G/Gst stable, and G00 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 Theq. The second result is an example of a group G definable in a non-distal, NIP theory for which G=G00 but Gst is not an intersection of definable groups. Our example is a saturated extension of (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.

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Full text arXiv 2110.02614: pdf, ps.