Benjamin D. Miller

The graph-theoretic approach to descriptive set theory

We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.

Descriptive Kakutani equivalence

We consider a descriptive set-theoretic analog of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two aperiodic Borel automorphisms of uncountable Polish spaces. Using this, we classify all Borel R-flows up to C∞-time-change isomorphism. We then extend the notion of descriptive Kakutani equivalence to all (not necessarily injective) Borel functions, and provide a variety of results leading towards a complete classification. The main technical tools are a series of Glimm-Effros and Dougherty-JacksonKechris-style embedding theorems.

The Borel cardinality of Lascar strong types

We show that if the restriction of the Lascar equivalence relation to a KP-strong type is non-trivial, then it is non-smooth (when viewed as a Borel equivalence relation on an appropriate space of types).

The existence of measures of a given cocycle, II: probability measures

Given a Polish space X , a countable Borel equivalence relation E on X , and a Borel cocycle ρ : E → (0,∞), we characterize the circumstances under which there is a probability measure μ on X such that ρ(φ−1(x), x) = [d(φ∗μ)/dμ](x) μ-almost everywhere, for every Borel injection φ whose graph is contained in E.

Ends of graphed equivalence relations, II

Given a graphing of a countable Borel equivalence relation on a Polish space, we show that if there is a Borel way of selecting a non-empty closed set of countably many ends from each -component, then there is a Borel way of selecting an end or line from each -component. Our method yields also Glimm-Effros style dichotomies which characterize the circumstances under which: (1) there is a Borel way of selecting a point or end from each -component; and (2) there is a Borel way of selecting a point, end or line from each -component.

An antibasis result for graphs of infinite Borel chromatic number

A graph on a set X is a symmetric, irreflexive subset of X ×X. For a graph G on X, we let degG(x) = |{y ∈ X : (x, y) ∈ G}|. If degG(x) is countable for all x ∈ X we say that G is locally countable. If, moreover, degG(x) is finite for all x ∈ X, we say that G is locally finite. The restriction of G to a set A ⊆ X, denoted by G|A, is simply G ∩ (A × A). A set A ⊆ X is (G)independent if G|A is empty. A κ-coloring of G is a function c : X → κ such that c−1(i) is independent for each i ∈ κ. The chromatic number of G, χ(G), is the least cardinal κ for which there exists a κ-coloring of G. Analogously, the Borel chromatic number, χB(G), of a graph on a standard Borel space X is the least cardinality of a Polish space Y for which there is a Borel function c : X → Y with c−1(y) a G-independent set for each y ∈ Y . For a graph G, let EG denote the equivalence relation generated by G. The classes of EG are called the connected components of G, and G is connected if EG has only one class. We say that G has indecomposably infinite Borel chromatic number if there is no way of partitioning the underlying space into countably many Borel EG-invariant sets such that the restriction of G to each has finite Borel chromatic number. We identify Ramsey space [N]N with the collection of increasing sequences of natural numbers. We then define the unilateral shift, s : [N]N → [N]N, by

BASIS THEOREMS FOR NON-POTENTIALLY CLOSED SETS AND GRAPHS OF UNCOUNTABLE BOREL CHROMATIC NUMBER

We show that there is an antichain basis for neither (1) the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor (2) the class of analytic graphs of uncountable Borel chromatic number under Borel reducibility.

Isomorphism of Borel full groups

Suppose that G and H are Polish groups which act in a Borel fashion on Polish spaces X and Y. Let E_G^X and E_H^Y denote the corresponding orbit equivalence relations, and [G] and [H] the corresponding Borel full groups. Modulo the obvious counterexamples, we show that [G] ≅ [H] ⇔ E_G^X ≅_B E_H^Y

INCOMPARABLE TREEABLE EQUIVALENCE RELATIONS

We establish Hjorths theorem that there is a family of continuum-many pairwise strongly incomparable free actions of free groups, and therefore a family of continuum-many pairwise incomparable treeable equivalence relations.

An embedding theorem of 𝔼0 with model theoretic applications

We provide a new criterion for embedding 𝔼0, and apply it to equivalence relations in model theory. This generalize the results of the authors and Pierre Simon on the Borel cardinality of Lascar strong types equality, and Newelskis results about pseudo Fσ groups.