Joshua Frisch, Alexander Kechris, Forte Shinko
Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.
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We consider countable Borel equivalence relations on quotient Borel spaces. We prove a generalization of the Feldman-Moore representation theorem, but provide some examples showing that other very simple properties of countable equivalence relations on standard Borel spaces may fail in the context of nonsmooth quotients.
The Borel complexity of conjugacy for Cantor minimal systems
本文证明 Cantor 极小系统共轭的 Borel 复杂度与可数图同构相同，通过编码 pro-有限群和逆向极限构造回答 Foreman 问题。
We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every $d > 1$ there is a $d$-regular acyclic hyperfinite Borel bipartite graph with no Borel perfect matching. These techniques also gi…
Decompositions and measures on countable Borel equivalence relations
本文证明可数Borel等价关系的测度论遍历分解可由生成它的群作用的拓扑遍历分解实现，并应用于基数代数与弱等价性研究。
We show that for any Polish group $G$ and any countable normal subgroup $Γ\triangleleft G$, the coset equivalence relation $G/Γ$ is a hyperfinite Borel equivalence relation. In particular, the outer automorphism group of any countable group is hyperfinite.
We study groups which satisfy Gardner's equidecomposition conjecture for uniformly distributed sets. We prove that an amenable group has this property if and only if it does not admit $(\mathbb{Z}/2\mathbb{Z}) *(\mathbb{Z}/2\mathbb{Z})$ as a quotient by a finite subgroup. Our technical contribution is an algorithm for rounding Borel flows for actions of amenable groups.
We introduce a notion of relative primeness for equivalence relations, strengthening the notion of non-reducibility, and show for many standard benchmark equivalence relations that non-reducibility may be strengthened to relative primeness. We introduce several analogues of cardinal properties for Borel equivalence relations, including the notion of a prime equivalence relation and Borel partition properties on quotient spaces. In particular, we introduce a notion of Borel weak compactness, and characterize partition properties for the equivalence relations ${\mathbb F}_2$ and ${\mathbb E}_1$.…
The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related spaces such as the Lascar group, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well as some new examples. The third is to explore notions of definable map, embedding and isomorphism between these and related quotient objects. The motivation for writing this paper is the recent discovery, via…
We show that the enveloping space $X_G$ of a partial action of a Polish group $G$ on a Polish space $X$ is a standard Borel space, that is to say, there is a topology $τ$ on $X_G$ such that $(X_G, τ)$ is Polish and the quotient Borel structure on $X_G$ is equal to $Borel(X_G,τ)$. To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught's transform are valid for partial actions of groups.
Reductions and necessary conditions for tall Borel Ramsey ideals
研究高博雷尔理想是否具拉姆齐性质这一开放问题，通过主要定理给出不存在可数局部解读等结论，还证明弱选择性$q^+$理想相关性质，提出博雷尔性是否强制可数解读的问题及后果，主要问题仍待解决。
No Countable Basis for Borel Directed Graphs of Dichromatic Number at Least Three
本文证明了Borel有向图中顶点可被分成两个Borel无环集的图类为Σ¹₂完全集，表明判断Borel有向图的二色数至少为三是一个Π¹₂完全问题。
In this paper we study the Borel structure of the space of left-orderings $\mathrm{LO}(G)$ of a group $G$ modulo the natural conjugacy action, and by using tools from descriptive set theory we find many examples of countable left-orderable groups such that the quotient space $\mathrm{LO}(G)/G$ is not standard. This answers a question of Deroin, Navas, and Rivas. We also prove that the countable Borel equivalence relation induced from the conjugacy action of $\mathbb{F}_{2}$ on $\mathrm{LO}(\mathbb{F}_{2})$ is universal, and leverage this result to provide many other examples of countable left-…
We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_σ$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $\emptyset$. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Andand Pillay, which says that for any strong type defined on a single complete type over $\emptyset$, smoothness is equivalent to type-definability. We also explai…
CONTENTS OF THE ISSUE: Hurewicz-like tests for Borel subsets of the plane; Ordered Spaces, Metric Preimages, and Function Algebras; On the independence of a generalized statement of Egoroff's theorem from ZFC, after T. Weiss; Forty annotated questions about large topological groups; Strong compactness and a partition property; Countable Borel equivalence relations and quotient Borel spaces; Decisive creatures and large continuum; Models of real-valued measurability; Hausdorff ultrafilters; Block combinatorics; Maharam's problem; Universality of uniform Eberlein compacta; Linearly ordered compa…
We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are "locally" quotients of compact Polish groups. It follows that they …
Solution to Dixmier's Problem about spectra of C*-algebras
本文解决了Dixmier关于单C*-代数谱的1967年问题，证明了在某些单C*-代数（如CAR代数与SL_3(Z)的约化交叉积）之间不存在Borel可定义的单射，并推广到可数amenable群与自由群的酉对偶，同时给出核非I型代数谱的Borel可定义双射存在性。
We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class $\bmΔ^1_2$ in a satisfying way.
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