Hyperdefinability of the Lie model for approximate subgroups

TL;DR

This paper investigates the definability of the locally compact group in the Lie model theorem for approximate groups, showing it is isomorphic to a relatively hyperdefinable locally compact group, thus deepening understanding of the model's structure.

Contribution

It demonstrates that the locally compact group in the Lie model is isomorphic to a relatively hyperdefinable group, enhancing the model's definability properties.

Findings

The locally compact group is relatively hyperdefinable.

The isomorphism preserves topological group structure.

Deepens understanding of the Lie model's definability.

Abstract

Hrushovski proved the Lie model theorem in full generality with model theoretic methods. The theorem states that for every approximate group there exists a generalized definable locally compact model, which, simplifying, is a quasi-homomorphism from the group generated by the approximate subgroup to a locally compact group with some particular properties. Pillay and Krupinski proved the same theorem using topological dynamics on a locally compact type space. In this paper we study the definability of the locally compact group image of the quasihomomorphism in this second proof. We show that it is isomorphic as a topological group to a relatively hyperdefinable locally compact group.