# Borel reductions of profinite actions of SLn(Z)

*Annals of pure and applied logic*, 2010. (arχiv | journal)

*Abstract*: 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 $3\leq m<n$ 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 $n$.