Looking for the lost torus

TL;DR

This paper classifies groups definable in coloured fields created via Hrushovski amalgamation, revealing their structure and showing that simple groups are algebraic, with specific descriptions in green and red fields.

Contribution

It provides a detailed classification of definable groups in coloured fields, highlighting their algebraic nature and structure in the context of Hrushovski constructions.

Findings

Groups in bad green fields are isogenous to quotients of algebraic subgroups by green points.

Subgroups of algebraic groups in coloured fields are extensions involving coloured points.

Simple groups in coloured fields are algebraic.

Abstract

We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of an algebraic group by a Cartesian power of the group of green elements. A definable subgroup of an algebraic group in the green or red field is an extension of the coloured points of a multiplicative or additive algebraic group by an algebraic group. In particular, a simple group in a coloured field is algebraic.