On the Complexity of the Isomorphism Relation for Finitely Generated Groups

Working within the framework of descriptive set theory, we show 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 on two generators. The proofs are group-theoretic, and we refer to descriptive set theory only for the relevant definitions and for motivation for the results.

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