Essential countability of treeable equivalence relations

Dominique Lecomte (1) , John D. Clemens (2) , Benjamin D. Miller (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 Institut für mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik

We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E is essentially countable if and only if there is no continuous embedding of E1 into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.

Mots clés: dichotomy theorem; essentially countable; essentially hyperfinite; hypersmooth

Domaines: Logique [math.LO]

Dominique Lecomte, John D. Clemens, Benjamin D. Miller. Essential countability of treeable equivalence relations. 2014. ⟨hal-01052451⟩