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3 Fundamenta Mathematicae

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Mixing actions of groups with the Haagerup approximation property

100%

Hjorth G.

Fundamenta Mathematicae

2009

tom 203

nr 1

47-56

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

100%

Hjorth G.

Fundamenta Mathematicae

2002

tom 175

nr 3

241-257

The isomorphism relation on countable torsion free abelian groups is non-Borel.

Recent developments in the theory of Borel reducibility

63%

Hjorth G. , Kechris A.

nr 1-2

21-52

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 Borel equivalence relation or else E₃ is reducible to E. We also survey a number of results and conjectures concerning the global structure of reducibility on Borel equivalence relations.