Isomorphism of finitely generated solvable groups is weakly universal

Abstract

We show that the isomorphism relation for finitely generated solvable groups of class 3 is a weakly universal countable Borel equivalence relation. This improves on previous results. The proof uses a modification of a construction of Neumann and Neumann. Elementary arguments show that isomorphism of finitely generated metabelian or nilpotent groups can not achieve this Borel complexity. In this sense the result is sharp, though it remains open whether the relation is in fact universal.

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