Borel reductions of profinite actions of

Greg Hjorth and Simon Thomas proved that the classification problem for torsion-free abelian groups of finite rank strictly increases in complexity with the rank. Subsequently, Thomas proved that the complexities of the classification problems for p -local torsion-free abelian groups of fixed rank n are pairwise incomparable as p varies. We prove that if 3 ≤ 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 .