Related papers: New jump operators on equivalence relations

This paper deals with countable products of countable Borel equivalence relations and equivalence relations "just above" those in the Borel reducibility hierarchy. We show that if $E$ is strongly ergodic with respect to $\mu$ then…

Let $\Gamma$ and $\Delta$ be sufficiently distinct countable groups. We show that there is an orbit equivalence relation $E$, induced by an action of the Polish wreath product group $\Gamma\wr\Gamma$, so that $E$ is generically $F$-ergodic…

We prove that for a countable discrete group $\Gamma$ containing a copy of the free group $\F_n$, for some $2\leq n\leq\infty$, as a normal subgroup, the equivalence relations of conjugacy, orbit equivalence and von Neumann equivalence of…

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence…

Recently, a surprising connection between algorithmic learning of algebraic structures and descriptive set theory has emerged. Following this line of research, we define the learning power of an equivalence relation $E$ on a topological…

A standard tool for classifying the complexity of equivalence relations on $\omega$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce…

We define a new variety of loops we call $\Gamma$-loops. After showing $\Gamma$-loops are power associative, our main goal will be showing a categorical isomorphism between Bruck loops of odd order and $\Gamma$-loops of odd order. Once this…

We study equivalence relations $\mathcal R(\Gamma\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $\Gamma\curvearrowright G$ is free and…

Building on work of Popa, Ioana, and Epstein--T\"{o}rnquist, we show that, for every nonamenable countable discrete group $\Gamma$, the relations of conjugacy, orbit equivalence, and von Neumann equivalence of free ergodic (or weak mixing)…

Fix $n=1,2,3,\dots$ or $n=\omega$. We prove a dichotomy for Borel homomorphisms from the $n$-th Friedman-Stanley jump $=^{+n}$ to an equivalence relation $E$ which is classifiable by countable structures: if there is no reduction from…

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones {\it et al}. More precisely, we show some jump and variational inequalities for the families $\mathcal…

We introduce an equivalence relation on the global class of morphisms of a category that extends several classical notions of equivalence in mathematics. We show that the standard group-action equivalence is a special case of our framework.…

We introduce the notion of $(\Gamma,E)$-determinacy for $\Gamma$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where…

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…

We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is…

We develop new tools to analyze the complexity of the conjugacy equivalence relation $E_\mathsf{lo}(G)$, whenever $G$ is a left-orderable group. Our methods are used to demonstrate non-smoothness of $E_\mathsf{lo}(G)$ for certain groups $G$…

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\mu$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group…

For a given group $G$, it is natural to ask whether one can classify all isometric $G$-actions on Gromov hyperbolic spaces. We propose a formalization of this problem utilizing the complexity theory of Borel equivalence relations. In this…

We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an $\omega$-Erdos cardinal, we determine which of these theories are Borel complete. We develop machinery,…

We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets…