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/