If G is a countable group containing a copy of F₂ then the conjugacy equivalence relation on subgroups of G attains the maximal possible complexity.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-3-1, author = {Alessandro Andretta and Riccardo Camerlo and Greg Hjorth}, title = {Conjugacy equivalence relation on subgroups}, journal = {Fundamenta Mathematicae}, volume = {167}, year = {2001}, pages = {189-212}, zbl = {0978.03037}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-3-1} }
Alessandro Andretta; Riccardo Camerlo; Greg Hjorth. Conjugacy equivalence relation on subgroups. Fundamenta Mathematicae, Tome 167 (2001) pp. 189-212. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-3-1/