Su Gao

Invariant descriptive set theory

Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets Coanalytic sets and pi 1/1 sets The Gandy-Harrington topology Polish Groups Metrics on topological groups Polish groups Continuity of homomorphisms The permutation group S Universal Polish groups The Graev metric groups Polish Group Actions Polish G-spaces The Vaught transforms Borel G-spaces Orbit equivalence relations Extensions of Polish group actions The logic actions Finer Polish Topologies Strong Choquet spaces Change of topology Finer topologies on Polish G-spaces Topological realization of Borel G-spaces Theory of Equivalence Relations Borel Reducibility Borel reductions Faithful Borel reductions Perfect set theorems for equivalence relations Smooth equivalence relations The Glimm-Effros Dichotomy The equivalence relation E0 Orbit equivalence relations embedding E0 The Harrington-Kechris-Louveau theorem Consequences of the Glimm-Effros dichotomy Actions of cli Polish groups Countable Borel Equivalence Relations Generalities of countable Borel equivalence relations Hyperfinite equivalence relations Universal countable Borel equivalence relations Amenable groups and amenable equivalence relations Actions of locally compact Polish groups Borel Equivalence Relations Hypersmooth equivalence relations Borel orbit equivalence relations A jump operator for Borel equivalence relations Examples of Fsigma equivalence relations Examples of pi 0/3 equivalence relations Analytic Equivalence Relations The Burgess trichotomy theorem Definable reductions among analytic equivalence relations Actions of standard Borel groups Wild Polish groups The topological Vaught conjecture Turbulent Actions of Polish Groups Homomorphisms and generic ergodicity Local orbits of Polish group actions Turbulent and generically turbulent actions The Hjorth turbulence theorem Examples of turbulence Orbit equivalence relations and E1 Countable Model Theory Polish Topologies of Infinitary Logic A review of first-order logic Model theory of infinitary logic Invariant Borel classes of countable models Polish topologies generated by countable fragments Atomic models and Gdelta-orbits The Scott Analysis Elements of the Scott analysis Borel approximations of isomorphism relations The Scott rank and computable ordinals A topological variation of the Scott analysis Sharp analysis of S -orbits Natural Classes of Countable Models Countable graphs Countable trees Countable linear orderings Countable groups Applications to Classification Problems Classification by Example: Polish Metric Spaces Standard Borel structures on hyperspaces Classification versus nonclassification Measurement of complexity Classification notions Summary of Benchmark Equivalence Relations Classification problems up to essential countability A roadmap of Borel equivalence relations Orbit equivalence relations General SIGMA 1/1 equivalence relations Beyond analyticity Appendix: Proofs about the Gandy-Harrington Topology The Gandy basis theorem The Gandy-Harrington topology on Xlow References Index

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C∗-algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

Computably Enumerable Equivalence Relations

We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.

A coloring property for countable groups

Motivated by research on hyperfinite equivalence relations we define a coloring property for countable groups. We prove that every countable group has the coloring property. This implies a compactness theorem for closed complete sections of the free part of the shift action of G on 2 G . Our theorems generalize known results about Z.

Group Colorings and Bernoulli Subflows

In this paper we study the dynamics of Bernoulli flows and their subflows over general countable groups from the symbolic and topological perspectives. We study free subflows (subflows in which every point has trivial stabilizer), minimal subflows, disjointness of subflows, the problem of classifying subflows up to topological conjugacy, and the differences in dynamical behavior between pairs of points which disagree on finitely many coordinates. We call a point hyper aperiodic if the closure of its orbit is a free subflow and we call it minimal if the closure of its orbit is a minimal subflow. We prove that the set of all (minimal) hyper aperiodic points is always dense but also meager and null. By employing notions and ideas from descriptive set theory, we study the complexity of the sets of hyper aperiodic points and of minimal points and completely determine their descriptive complexity. In doing this we introduce a new notion of countable flecc groups and study their properties. We obtain a dichotomy for the complexity of classifying free subflows up to topological conjugacy. For locally finite groups the topological conjugacy relation for all (free) subflows is hyperfinite and nonsmooth. For nonlocally finite groups the relation is Borel bireducible with the universal countable Borel equivalence relation. A primary focus of the paper is to develop constructive methods for the notions studied. To construct hyper aperiodic points, a fundamental method of construction of multi-layer marker structures is developed with great generality. Variations of the fundamental method are used in many proofs in the paper, and we expect them to be useful more broadly in geometric group theory. As a special case of such marker structures, we study the notion of ccc groups and prove the ccc-ness for countable nilpotent, polycyclic, residually finite, locally finite groups and for free products.

Resolvable maps preserve complete metrizability

Let X be a Polish space, Y a separable metrizable space, and f : X → Y a continuous surjection. We prove that if the image under f of every open set or every closed set is resolvable, then Y is Polish. This generalizes similar results by Sierpinski, Vainstain, and Ostrovsky.

Some applications of the Adams-Kechris technique

We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every Σ_1^1 equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence relations and derive analogous results for the classification problem of Borel automorphisms.

Complexity Ranks of Countable Models

We define some variations of the Scott rank for countable models and obtain some inequalities involving the ranks. For mono-unary algebras we prove that the game rank of any subtree does not exceed the game rank of the whole model. However, similar questions about linear orders remain unresolved.

Some Dichotomy Theorems for Isomorphism Relations of Countable Models

Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an L ω 1 ω -interpretability theory. We also give a survey of this theory.

Coding Subset Shift by Subgroup Conjugacy

We present a Borel reduction from a subset shift equivalence relation of a countable group to a subgroup conjugacy relation of a free product. The technique gives a much shorter proof of an earlier result of Thomas and Velickovic.