Borel reductions of profinite actions of SLn(Z)

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 complexity 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.