On the complexity of the classification problem for torsion-free Abelian groups of finite rank (Q2778661)

It has long been known (Baer, 1937) that the set of types of torsion-free Abelian groups of rank 1 is a complete set of invariants, but there is currently no satisfactory system of complete invariants for torsion-free Abelian groups of rank \(n\geq 2\). In this paper, Thomas offers some explanations for why this might be so using descriptive set theory developed primarily by Friedman-Stanley and Hjorth-Kechris.NEWLINENEWLINENEWLINELet \(E\) and \(F\) be Borel equivalence relations on the Borel spaces \(X\) and \(Y\) respectively, that is \(E\subseteq X^2\) is a Borel subset of \(X^2\), and similarly for \(F\). We write \(E\leq_BF\) if there exists a Borel function \(f\colon X\to Y\) such that \(xEy\) iff \(f(x)Ff(y)\). We write \(E<_BF\) if \(E\leq_BF\) and \(F\not\leq_BE\). It is known that the set of torsion-free Abelian groups of rank \(\leq n\) can be identified with the set \(S(\mathbb{Q}^n)\) of additive subgroups of \(\mathbb{Q}^n\) and that the isomorphism relation on \(S(\mathbb{Q}^n)\), denoted \(\cong_n\), is a countable Borel equivalence relation. For Thomas, the question of the difficulty of the classification problem for torsion-free Abelian groups of rank \(n\geq 2\) can be answered by determining whether \((\cong_1)<_B(\cong_n)\) when \(n\geq 2\). This was solved positively by Hjorth in 1999, and Thomas includes an outline of the proof in this paper.NEWLINENEWLINENEWLINEOf related interest, in 2000 Adams and Kechris used Zimmer's superrigidity theory to prove that \((\cong_n^*)<_B(\cong_{n+1}^*)\) for \(n\geq 1\) where \(\cong_n^*\) is the restriction of the isomorphism relation to the Borel subset \(S^*(\mathbb{Q}^n)\) consisting of the rigid groups \(A\in S(\mathbb{Q}^n)\). A sketch of the proof for this result is included.NEWLINENEWLINENEWLINEFinally, Thomas himself has recently (2000) shown that \((\cong_{n+1}^*)\not\leq_B(\cong_n)\) for all \(n\geq 1\) which then implies that \((\cong_n)<_B(\cong_{n+1})\) for all \(n\geq 1\). A sketch of this proof is included as well.