Steve Jackson

Countable borel equivalence relations

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyper-finite, amenable, treeable and universal equivalence relations.

The structure of hyperfinite Borel equivalence relations

We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.

Structural Consequences of AD

In this chapter we survey recent advances in descriptive set theory, starting (roughly) from where Moschovakis’ book (1980) ends. Our survey is not intended to be complete, but focuses mainly on the structural consequences of determinacy for the model L(ℝ), including the important case of the projective sets. By “structural” we are referring to the combinatorial theory of the pointclasses (for example, the scale property which in some sense describes the structure of the set) as well as the cardinal structure up to the natural ordinal associated with these pointclasses. This might include determining their cofinalities, partition properties, and so forth.

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.

On a lattice problem of H. Steinhaus

Sometime in the 1950s, Steinhaus posed the following problem. Do there exist two sets A and S in the plane such that every set congruent to A has exactly one point in common with S? The trivial case where one of the sets is the plane and the other consists of a single point is ruled out. The first appearance of this problem in the literature seems to be in a 1958 paper of Sierpinski [14]. In this paper, he showed the answer is yes, a result later rediscovered by Erdos [5]. Of course, there are many variants of this problem. For example, one could specify the set A. In this direction, Komjaith showed that such a set exists if A = Z, the set of all integers [13]. Steinhaus also asked about the specific case where A = 2. The first reference to this problem also seems to be Sierpinskis 1958 paper where he mentions that in this case there is no set S which is bounded and open or else bounded and closed. This specific problem has been widely noted (see e.g. [3, 4]), but has remained unsolved until now. In this paper we answer this question in the affirmative:

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.

Sets meeting isometric copies of the lattice Z2 in exactly one point.

The construction of a subset S of ℝ2 such that each isometric copy of ℤ2 (the lattice points in the plane) meets S in exactly one point is indicated. This provides a positive answer to a problem of H. Steinhaus [Sierpiński, W. (1958) Fund. Math. 46, 191–194].

Følner tilings for actions of amenable groups

We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G (“shapes”) with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz–Huczek–Zhang tiling theorem for countable amenable groups and strengthens the Ornstein–Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is

Supercompactness within the projective hierarchy

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