# The complexity of the embeddability relation between torsion-free abelian groups of uncountable size

## TL;DR

This paper demonstrates the high complexity of the embeddability relation among uncountably large torsion-free abelian groups by reducing it from graph embeddability, establishing its status as a complete analytic quasi-order.

## Contribution

It proves that embeddability among uncountable torsion-free abelian groups is as complex as graph embeddability, extending to modules over certain rings, and establishes their completeness in the analytic hierarchy.

## Key findings

- Embeddability on uncountable torsion-free abelian groups is a complete analytic quasi-order.
- Reductions from graph embeddability to group embeddability are established.
- The results extend to modules over specific rings, showing broad applicability.

## Abstract

We prove that for every uncountable cardinal $\kappa$ such that $\kappa^{<\kappa}=\kappa$, the quasi-order of embeddability on the $\kappa$-space of $\kappa$-sized graphs Borel reduces to the embeddability on the $\kappa$-space of $\kappa$-sized torsion-free abelian groups. Then we use the same techniques to prove that the former Borel reduces to the embeddability on the $\kappa$-space of $\kappa$-sized $R$-modules, for every $\mathbb{S}$-cotorsion-free ring $R$ of cardinality less than the continuum. As a consequence we get that all the previous are complete $\boldsymbol{\Sigma}^1_1$ quasi-orders.