Related papers: Countable Borel treeable equivalence relations are…
This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence…
A variation of the Scott analysis of countable structures is applied to actions of non-Archimedean TSI Polish groups acting continuously on a Polish spaces. We give results on the potential Borel complexity spectrum of such groups, and…
We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an…
A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we show that this question has a…
We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially…
Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We…
We develop a unified framework for locating natural properties of algebraic and analytic structures within the Borel hierarchy. Objects are presented as quotients of a universal generator and definability is read directly from the quotient…
In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are separable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum…
We prove that orbit equivalence relations (ERs, for brevity) of generically turbulent Polish actions are not Borel reducible to ER s of a family which includes Polish actions of S_\infty, the group of all permutations of N, and is closed…
In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the…
In recent years, much work has been done to measure and compare the complexity of orbit equivalence relations, especially for certain classes of Polish groups. We start by introducing some language to organize this previous work, namely the…
We consider reducibility of equivalence relations (ERs, for brevity), in a nonstandard domain, in terms of the Borel reducibility and the countably determined (CD, for brevity) reducibility. This reveals phenomena partially analogous to…
We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect…
The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group $G$, let $E(G)$ be the…
An uncountable $\aleph_1$-free group cannot admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed, for example, by the Baer-Specker group $\mathbb{Z}^\omega$; more strongly, $\mathbb{Z}^\omega$ is…
We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a…
Let $(X,\tau)$ be a Polish space with Borel probability measure $\mu,$ and $G$ a locally finite one-ended Borel graph on $X.$ We show that $G$ admits a Borel one-ended spanning tree generically. If $G$ is induced by a free Borel action of…
A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we prove that this question…
We consider various notions of equivalence in the space of bounded operators on a Hilbert space, in particular modulo finite rank, modulo Schatten $p$-class, and modulo compact. Using Hjorth's theory of turbulence, the latter two are shown…
Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is…