John Clemens: Treeable equivalence relations and descriptive complexity

Monday, May 14, 2012, 16.30

Speaker: John Clemens (University of Münster)

Title: Treeable equivalence relations and descriptive complexity

Abstract:

The property of treeability has been well-studied in the context of countable Borel equivalence relations, but has been little studied up to now in the uncountable case. I will discuss a family of treeable equivalence relations which provides new insights into this area as well as the area of potential descriptive complexity. In particular, we show that the collection of treeable Borel equivalence relations is unbounded in the Borel-reducibility hierarchy. Additionally, a generalization of the Kechris-Louveau dichotomoy for E_1 allows us to show that for every Borel equivalence relation which is not essentially hyperfinite we may find equivalence relations of arbitrarily high descriptive complexity with which it is incomparable under Borel reducibility. This is joint work with Dominique Lecomte and Ben Miller.