On the Complexity of the Isomorphism Relation for Finitely Generated Groups,

Confirming a conjecture of G. Hjorth and A. Kechris (1996, Ann. Pure Appl. Logic82, 221–272) we prove that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group F2 on two generators.