Related papers: Borel structures on the space of left-orderings

We show that a {\it Borel} action of a Polish group on a standard Borel space is Borel isomorphic to a {\it continuous} action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of…

We show that given a monadically stable theory $T$, a sufficiently saturated $\mathbf M \models T$, and a coherent system of probability measures on the $\sigma$-algebras generated by parameter-definable sets of $\mathbf M$ in each…

We extend Furstenberg's structure theorem to any standard Borel $G$-space, where $G$ is any locally compact second countable Noetherian module over a syndetic ring.

We compute the Borel equivariant cohomology ring of the left $K$-action on a homogeneous space $G/H$, where $G$ is a connected Lie group, $H$ and $K$ are closed, connected subgroups and $2$ and the torsion primes of the Lie groups are units…

We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson…

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…

We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set. We say that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager…

An algebraic left Kan extension is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of…

Let $E\subseteq F$ and $E'\subseteq F'$ be Borel equivalence relations on the standard Borel spaces $X$ and $Y$, respectively. The pair $(E,F)$ is simultaneously Borel reducible to the pair $(E',F')$ if there is a Borel function $f:X\to Y$…

In this paper, a notion of Schauder equivalence relation $\mathbb R^\mathbb N/L$ is introduced, where $L$ is a linear subspace of $\mathbb R^\mathbb N$ and the unit vectors form a Schauder basis of $L$. The main theorem is to show that the…

We investigate $\mathcal F$-Borel topological spaces. We focus on finding out how a~complexity of a~space depends on where the~space is embedded. Of a~particular interest is the~problem of determining whether a~complexity of given space $X$…

We study generalized sums of linear orders. These are binary operations that, given linear orders $A$ and $B$, return an order $A \oplus B$ that can be decomposed as an isomorphic copy of $A$ interleaved with a copy of $B$. We show that…

Gao and Jackson showed that any countable Borel equivalence relation (CBER) induced by a countable abelian Polish group is hyperfinite. This prompted Hjorth to ask if this is in fact true for all CBERs classifiable by (uncountable) abelian…

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…

Boykin and Jackson recently introduced a property of countable Borel equivalence relations called Borel boundedness, which they showed is closely related to the union problem for hyperfinite equivalence relations. In this paper, we…

We study Borel homomorphisms $\theta : G\rightarrow H$ for arbitrary locally compact second countable groups $G$ and $H$ for which the measure $$\theta_*(\mu )(\alpha )=\mu (\theta ^{-1}(\alpha ))\quad \text{for } \quad \alpha \subseteq H…

We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel…

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…

For $N \geq 2$, we study the structure of definable abelian group extensions of the additive group $(\mathbb{R}^N,+)$ by countable abelian (Borel) groups $G$. Given an extension $H$ of $(\mathbb{R}^N,+)$ by $G$, we measure the definability…

Let ${\cal S}(\mathcal{H})$ denote the set of all self-adjoint operators (not necessarily bounded) on a Hilbert space $\mathcal{H}$, which is the set of all physical quantities on a quantum system $\mathcal{H}$. We introduce a binary…