Greg Hjorth

Classification and orbit equivalence relations

An outline Definitions and technicalities Turbulence Classifying homeomorphisms Infinite dimensional group representations A generalized Scott analysis GE groups The dark side Beyond Borel Looking ahead Ordinals Notation Bibliography Index.

A converse to dye's theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F 2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the ≤ B ordering.

A lemma for cost attained

Abstract A treeable ergodic equivalence relation of integer cost is generated by a free action of the free group on the corresponding number of generators. Every countable treeable ergodic equivalence relation is induced by the free action of some countable group.

Analytic Equivalence Relations and Ulm-Type Classifications

Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.

Borel equivalence relations induced by actions of the symmetric group

We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π^0_3 iff the invariants are countable sets of reals, it is potentially Π^0_4 iff the invariants are countable sets of countable sets of reals, and so on. The proofs use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models.

New Dichotomies for Borel Equivalence Relations

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.

Actions by the classical Banach spaces

?0. Preface. The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space. For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models. This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that the Borel cardinalities of the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action. One of the first things to be addressed concerns the complexity of these equivalence relations. This question for B = 12 appears in [1].

From Automatic Structures to Borel Structures

We study the classes of Buchi and Rabin automatic structures. For Buchi (Rabin) automatic structures their domains consist of infinite strings (trees), and the basic relations, including the equality relation, and graphs of operations are recognized by Buchi (Rabin) automata. A Buchi (Rabin) automatic structure is injective if different infinite strings (trees) represent different elements of the structure. The first part of the paper is devoted to understanding the automata- theoretic content of the well-known Lowenheim-Skolem theorem in model theory. We provide automata-theoretic versions of Lowenheim-Skolem theorem for Rabin and Buchi automatic structures. In the second part, we address the following two well-known open problems in the theory of automatic structures: Does every Buchi automatic structure have an injective Buchi presentation? Does every Rabin automatic structure have an injective Rabin presentation? We provide examples of Buchi structures without injective Buchi and Rabin presentations. To answer these questions we introduce Borel structures and use some of the basic properties of Borel sets and isomorphisms. Finally, in the last part of the paper we study the isomorphism problem for Buchi automatic structures.

Orbit cardinals: On the effective cardinalities arising as quotient spaces of the formX/G whereG acts on a Polish spaceX

We prove an Ulm-type classification theorem for actions inL(ℝ), thereby answering a question of Becker and Kechris, and investigate the effective cardinalities which can be induced by various classes of Polish groups.

KNIGHT'S MODEL, ITS AUTOMORPHISM GROUP, AND CHARACTERIZING THE UNCOUNTABLE CARDINALS

We show that every ℵα (α<ω1) can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1 but whose automorphism group fails the topological Vaught conjecture on analytic sets. We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knights model.