Relative geometries

TL;DR

This paper explores geometric properties of structures relative to reducts, focusing on definability of groups and fields, and characterizes the structure of definable groups in relatively one-based and CM-trivial cases.

Contribution

It introduces a framework for analyzing geometric properties relative to reducts and characterizes definable groups in these contexts, including cases involving Hrushovski amalgamations.

Findings

Definable groups in relatively one-based structures are isogenous to subgroups of products of reduct-definable groups.

In relatively CM-trivial structures, definable groups admit homomorphisms with virtually central kernels into products of reduct-definable groups.

The analysis includes structures arising from Hrushovski amalgamations and field expansions by predicates.

Abstract

We start an analysis of geometric properties of a structure relative to a reduct. In particular, we look at definability of groups and fields in this context. In the relatively one-based case, every definable group is isogenous to a subgroup of a product of groups definable in the reducts. In the relatively CM-trivial case, which contains certain Hrushovski amalgamations (the fusion of two strongly minimal sets or the expansions of a field by a predicate), every definable group allows a homomorphism with virtually central kernel into a product of groups definable in the reducts.