The classification of torsion-free abelian groups of finite rank up to isomorphism and up to quasi-isomorphism

TL;DR

This paper investigates the complexity of classifying torsion-free abelian groups of finite rank, showing that isomorphism and quasi-isomorphism relations are incomparable in terms of Borel reducibility.

Contribution

It establishes the incomparability of isomorphism and quasi-isomorphism relations for p-local torsion-free abelian groups of rank at least 3.

Findings

Isomorphism and quasi-isomorphism are incomparable under Borel reducibility.

Focus on p-local torsion-free abelian groups of rank ≥ 3.

Provides insights into classification complexity of these groups.

Abstract

The isomorphism and quasi-isomorphism relations on the $p$-local torsion-free abelian groups of rank $n\geq3$ are incomparable with respect to Borel reducibility.