On the complexity of the classification problem for torsion-free abelian groups of rank two

Adams, S. R. &Kechris, A. S., Linear algebraic groups and countable Borel equivalence relations.J. Amer. Math. Soc., 13 (2000), 909–943. Adams, S. R. &Spatzier, R. J., Kazhdan groups, cocycles and trees.Amer. J. Math., 112 (1990), 271–287. Baer, R., Abelian groups without elements of finite order.Duke Math. J., 3 (1937), 68–122. Connes, A., Feldman, J. &Weiss, B., An amenable equivalence relation is generated by a single transformation.Ergodic Theory Dynam. Systems, 1 (1981), 431–450. Dougherty, R., Jackson, S. &Kechris, A. S., The structure of hyperfinite Borel equivalence relations.Trans. Amer. Math. Soc., 341 (1994), 193–225. Feldman, J. &Moore, C. C., Ergodic equivalence relations, cohomology and von Neumann algebras, I.Trans. Amer. Math. Soc., 234 (1977), 289–324. Fuchs, L.,Infinite Abelian Groups, Vol. II. Pure Appl. Math., 36-II. Academic Press, New York-London, 1973. Hjorth, G., Around nonclassifiability for countable torsion-free abelian groups, inAbelian Groups and Modules (Dublin, 1998), pp. 269–292. Trends Math., Birkhäuser, Basel, 1999. Hjorth, G. &Kechris, A. S., Borel equivalence relations and classifications of countable models.Ann. Pure Appl. Logic, 82 (1996), 221–272. Jackson, S., Kechris, A. S. &Louveau, A., Countable Borel equivalence relations.J. Math. Log., 2 (2002), 1–80. Kechris, A. S., Countable sections for locally compact group actions.Ergodic Theory Dynam. Systems, 12 (1992), 283–295. —, Amenable versus hyperfinite Borel equivalence relations.J. Symbolic Logic, 58 (1993), 894–907. —, On the classification problem for rank 2 torsion-free abelian groups.J. London Math. Soc. (2), 62 (2000), 437–450. Król, M.,The Automorphism Groups and Endomorphism Rings of Torsion-Free Abelian Groups of Rank Two. Dissertationes Math. (Rozprawy Mat.), 55. PWN, Warsaw, 1967. Kurosh, A. G., Primitive torsionsfreie abelsche Gruppen vom endlichen Range.Ann. of Math. (2), 38 (1937), 175–203. Malcev, A. I., Torsion-free abelian groups of finite rank.Mat. Sb. (N.S.), 4 (1938), 45–68 (Russian). Margulis, G. A.,Discrete Subgroups of Semisimple Lie Groups. Ergeb. Math. Grenzgeb. (3), 17. Springer-Verlag, Berlin, 1991. Pierce, R. S.,Associative Algebras. Graduate Texts in Math., 88. Springer-Verlag, New York-Berlin, 1982. Rotman, J. J.,An Introduction to Homological Algebra. Pure Appl. Math., 85. Academic Press, New York-London, 1979. Serre, J.-P.,Arbres, amalgames, SL 2. Astérisque, 46. Soc. Math. France, Paris, 1977. Thomas, S., The classification problem for torsion-free abelian groups of finite rank.J. Amer. Math. Soc., 16 (2003), 233–258. Thomas, S. &Velickovic, B., On the complexity of the isomorphism relation for finitely generated groups.J. Algebra, 217 (1999), 352–373. —, On the complexity of the isomorphism relation for fields of finite transcendence degree.J. Pure Appl. Algebra, 159 (2001), 347–363. Wagon, S.,The Banach-Tarski Paradox. Encyclopedia Math. Appl., 24. Cambridge Univ. Press, Cambridge, 1985. Zimmer, R.,Ergodic Theory and Semisimple Groups. Monographs Math., 81. Birkhäuser, Basel, 1984. —, Kazhdan groups acting on compact manifolds.Invent. Math., 75 (1984), 425–436. —, Groups generating transversals to semisimple Lie group actions.Israel J. Math., 73 (1991), 151–159.