The classification problem for torsion-free abelian groups of finite rank

TL;DR: In this paper, it was shown that the classification problem for the groups of rank n > 2 is intractible, and that there are 2' pairwise nonisomorphic groups up to isomorphism.

Abstract: In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-free abelian groups of finite rank n > 2. So it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n > 2. Of course, if we wish to show that the classification problem for the groups of rank n > 2 is intractible, it is not enough merely to prove that there are 2' such groups up to isomorphism. For there are 2' pairwise nonisomorphic groups of rank 1, and we have already pointed out that Baer has given a satisfactory classification for this class of groups. In this paper, following Friedman-Stanley [11] and Hjorth-Kechris [15], we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n > 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space Qf which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank 1 < r < n can be naturally identified with the set S(Qn) of all nontrivial additive subgroups of Qn. Notice that S(Qn) is a Borel subset of the Polish space ,p(Qn) of all subsets of Qn, and hence S(Qn) can be regarded as a standard Borel space; i.e. a Polish space equipped with its associated o-algebra of Borel subsets. (Here we are identifying p(Qn) with the space 2'Qn of all functions h: Qn {0, 1} equipped with the product topology.) Furthermore, the natural action of GLn(Q) on the vector space Qn induces a corresponding Borel action on S(Qn); and it is easily checked that if A, B C S(Qn), then A _V B iff there exists an element f E GLn(Q) such that p(A) = B. It follows that the isomorphism relation on S(Qn) is a countable Borel equivalence relation. (If X is a standard Borel space, then a Borel equivalence relation on X is an equivalence relation E C X2 which is a Borel subset of X2. The Borel equivalence relation E is said to be countable iff every E-equivalence class is countable.)