The isomorphism relation on countable torsion free abelian groups is non-Borel.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-3-2, author = {Greg Hjorth}, title = {The isomorphism relation on countable torsion free abelian groups}, journal = {Fundamenta Mathematicae}, volume = {173}, year = {2002}, pages = {241-257}, zbl = {1021.03042}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-3-2} }
Greg Hjorth. The isomorphism relation on countable torsion free abelian groups. Fundamenta Mathematicae, Tome 173 (2002) pp. 241-257. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm175-3-2/