On piecewise hyperdefinable groups

TL;DR

This paper extends key model-theoretic results from stable group theory to piecewise hyperdefinable sets, developing their structure and properties, including Lie models and stabilizer theorems.

Contribution

It generalizes and improves Hrushovski's results to a broader class of sets, providing a systematic study of their structure and topology.

Findings

Existence of Lie models for piecewise hyperdefinable groups

Development of a stabilizer theorem in this context

Analysis of the logic topologies of these sets

Abstract

The aim of this paper is to generalize and improve two of the main model-theoretic results of "Stable group theory and approximate subgroups" by E. Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the stabilizer theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies.