Borel reductions of profinite actions of SL(n,Z)

TL;DR

This paper investigates the complexity of classifying torsion-free abelian groups of various ranks and localizations, showing that these complexities are pairwise incomparable across different primes and ranks, revealing a rich structure in classification problems.

Contribution

It extends previous results by demonstrating that classification complexities for torsion-free abelian groups are pairwise incomparable across different primes and ranks, deepening understanding of their complexity landscape.

Findings

Complexity increases with rank for torsion-free abelian groups.

Classification problems for different primes and ranks are pairwise incomparable.

The results reveal a complex hierarchy in classification difficulties.

Abstract

Greg Hjorth and Simon Thomas proved that the classification problem for torsion-free abelian groups of finite rank \emph{strictly increases} in complexity with the rank. Subsequently, Thomas proved that the complexity of the classification problems for $p$-local torsion-free abelian groups of fixed rank $n$ are \emph{pairwise incomparable} as $p$ varies. We prove that if $3\leq m<n$ and $p,q$ are distinct primes, then the complexity of the classification problem for $p$-local torsion-free abelian groups of rank $m$ is again incomparable with that for $q$-local torsion-free abelian groups of rank $n$.