Abstract: In recent years, a major theme in descriptive set theory has been the study of the Borel complexity of naturally occurring classification problems. For example, Hjorth and Thomas have shown that the Borel complexity of the isomorphism problem for the torsion-free abelian groups of rank $n$ increases strictly with the rank $n$.
In this thesis, we present some new applications of the theory of countable Borel equivalence relations to various classification problems for the $p$-local torsion-free abelian groups of finite rank. Our main result is that when $n\geq3$, the isomorphism and quasi-isomorphism problems for the $p$-local torsion-free abelian groups of rank $n$ have incomparable Borel complexities. (Here two abelian groups $A$ and $B$ are said to be quasi-isomorphic if $A$ is abstractly commensurable with $B$.) We also introduce a new invariant, the divisible rank, for the class of $p$-local torsion-free abelian groups of finite rank; and we prove that if $n\geq3$ and $1\leq k\leq n-1$, then the isomorphism problems for the $p$-local torsion-free abelian groups of rank $n$ and divisible rank $k$ have incomparable Borel complexities as $k$ varies.
Our proofs rely on the framework developed by Adams and Kechris, whereby cocycle superrigidity results from measurable group theory are applied in the purely Borel setting. In particular, we make use of the recent cocycle superrigidity theorem, due to Ioana, for free ergodic profinite actions of Kazhdan groups.