On stable quotients

TL;DR

This paper addresses two problems related to maximal stable quotients of groups in NIP theories, proving a key equality in distal theories and providing a counterexample in non-distal NIP theories, with implications for stability characterizations.

Contribution

It proves that in distal theories, the stable quotient equals the smallest type-definable subgroup of bounded index, and constructs a non-distal NIP example where this equality fails.

Findings

In distal theories, G^{st} equals G^{00}.

Constructed a non-distal NIP example with G=G^{00} but G^{st} not an intersection of definable groups.

Provided characterizations of stability of hyperdefinable sets using continuous logic.

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.