Essential countability of treeable equivalence relations

TL;DR

This paper characterizes when treeable Borel equivalence relations are essentially countable, linking this property to the absence of continuous embeddings of E1, and provides new proofs and minimal examples in the theory.

Contribution

It establishes a dichotomy theorem for essential countability of treeable Borel equivalence relations and introduces novel techniques for classical results and minimal non-countable-to-one functions.

Findings

E is essentially countable iff no continuous embedding of E1 exists under certain conditions

Provides the first classical proof for hypersmooth equivalence relations

Identifies a minimal Borel function that is not essentially countable-to-one

Abstract

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.