When is an equivalance relation classifiable.
Mixing actions of groups with the Haagerup approximation property
A countable group Γ has the Haagerup approximation property if and only if the mixing actions are dense in the space of all actions of Γ.
The isomorphism relation on countable torsion free abelian groups
The isomorphism relation on countable torsion free abelian groups is non-Borel.
Recent developments in the theory of Borel reducibility
Let E₀ be the Vitali equivalence relation and E₃ the product of countably many copies of E₀. Two new dichotomy theorems for Borel equivalence relations are proved. First, for any Borel equivalence relation E that is (Borel) reducible to E₃, either E is reducible to E₀ or else E₃ is reducible to E. Second, if E is a Borel equivalence relation induced by a Borel action of a closed subgroup of the infinite symmetric group that admits an invariant metric, then either E is reducible to a countable...
Conjugacy equivalence relation on subgroups
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.
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