Groups definable in two orthogonal sorts

TL;DR

This paper investigates groups interpretable in a two-sorted structure, showing they form extensions of groups internal to each sort, with specific structural properties when one sort is superstable or o-minimal.

Contribution

It characterizes the structure of interpretable groups in two orthogonal sorts, especially under superstable and o-minimal conditions, revealing their extension and Lie group structures.

Findings

Groups are extensions of internal groups to each sort.

If one sort is superstable with definable Lascar rank, the group is an extension of an internal group by a definable subgroup.

In o-minimal cases, the group admits a Lie structure and the extension is a topological cover.

Abstract

This work can be thought as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover.