Related papers: Quotients by countable subgroups are hyperfinite

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 study cocycles of countable groups $\Gamma$ of Borel automorphisms of a standard Borel space $(X, \mathcal{B})$ taking values in a locally compact second countable group $G$. We prove that for a hyperfinite group $\Gamma$ the subgroup of…

We show that if there exists a countable Borel equivalence relation which is hyper-hyperfinite but not hyperfinite then the complexity of hyperfinite countable Borel equivalence relations is as high as possible, namely,…

We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel…

Let $G$ be a Polish group and let $H \leq G$ be a compact subgroup. We prove that there exists a Borel set $T \subset G$ which is simultaneously a complete set of coset representatives of left and right cosets, provided that a certain index…

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_\sigma$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to…

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based…

Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence…

Let $G$ be a group that is relatively hyperbolic with respect to a collection of subgroups $\{H_{\lambda}\}_{\lambda\in \Lambda}$. Suppose that $G$ is given by a finite relative presentation $\mathcal{P}$ with respect to this collection. We…

We prove that no quantifier-free formula in the language of group theory can define the $\aleph_1$-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of…

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…

Given a finitely generated group $G$ that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ is…

We prove that there exists a countable metrizable topological group $G$ such that every countable metrizable group is isomorphic to a quotient of $G$. The completion $H$ of $G$ is a Polish group such that every Polish group is isomorphic to…

It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group $S_\infty$ is continuous. It is…

Working in the framework of Borel reducibility, we study various notions of embeddability between groups. We prove that the embeddability between countable groups, the topological embeddability between (discrete) Polish groups, and the…

Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^\omega/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. The connected component of the identity of a Polish group $G$ is denoted by $G_0$.…

We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.

We show that if $G$ is a finitely generated group hyperbolic relative to a finite collection of subgroups $\mathcal{P}$, then the natural action of $G$ on the geodesic boundary of the associated relative Cayley graph induces a hyperfinite…

We show that every non-Archimedean Polish group $P$ is the outer automorphism group of a countable discrete group $G_P$. Moreover, our construction provides a Borel map $f$ from the Effros space of closed subgroups of the permutation group…

We show that if a group $G$ has a finite normal subgroup $L$ such that $G/L$ is hypercentral, then the index of the hypercenter of $G$ is bounded by a function of the order of $L$. This completes recent results generalizing classical…