Model-theoretic Tameness in finite extensions of groups

TL;DR

The paper demonstrates that certain finite-index extensions and subgroups of omega-stable groups can interpret any countable structure, showing their potential for model-theoretic complexity.

Contribution

It reveals that finite-index extensions and subgroups of omega-stable groups can be arbitrarily complex in a model-theoretic sense, contrary to expectations of tameness.

Findings

Finite-index extensions of omega-stable groups can interpret any countable structure.

Finite-index subgroups of omega-stable groups can also interpret any countable structure.

Omega-stable groups can have model-theoretic wildness in their finite extensions and subgroups.

Abstract

It is shown that finite-index extensions and finite-index subgroups of $\omega$-stable groups can be model-theoretically wild. More precisely, there exists an $\omega$-stable group $G$ such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of $G$ and in some finite-index subgroup of $G$.